Bifurcation for Minimal Surface Equation in Hyperbolic $3$-Manifolds

TL;DR

This paper investigates the existence, multiplicity, and asymptotic behavior of minimal surface immersions in hyperbolic 3-manifolds, revealing conditions for uniqueness, multiple solutions, and blow-up phenomena, with extensions to prescribed extrinsic curvature.

Contribution

It provides new criteria for solution uniqueness, analyzes blow-up behavior based on genus, and extends the theory to prescribed extrinsic curvature problems.

Findings

Unique solutions determined by conformal structure and second fundamental form

Multiple solutions can occur depending on geometric conditions

Blow-up behavior varies with the genus of the surface

Abstract

Initiated by the work of Uhlenbeck in late 1970s, we study questions about the existence, multiplicity and asymptotic behavior for minimal immersions of closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions. In particular, we determine when exactly the solution is unique and when multiple solutions appear. Moreover, we analyze in detail the asymptotic behavior of the solutions when (and how) blowing up might occur. Interestingly the blow-up analysis exhibit different behaviors when the surface is of genus two or greater. Furthermore, we extend this program to consider similar problems where the total extrinsic curvature is prescribed and we prove an existence result.

Equations

$$g_0=e^{2u}{g}_{\sigma }.$$

$$g_{\sigma }=e^{2u_{\sigma }}dzd\bar{z},\text{ and }{g}_0=e^{2v}dzd\bar{z},$$

$$\text{II}=h_{11}(dx{)}^2+2h_{12}dxdy+h_{22}(dy{)}^2,$$

$$R_{ij\kappa \ell }=-(g_{i\kappa }{g}_{j\ell }-g_{i\ell }{g}_{j\kappa }).$$

$$R_{i3\kappa 3}=-g_{i\kappa }.$$

$$\left\{\begin{array}[]{cl}g_{i\kappa}(z,0)=(g_{0})_{i\kappa}(z)\vspace{5pt}\\ \frac{1}{2}\frac{\partial}{\partial r}g_{i\kappa}(z,0)=h_{i\kappa}(z),\ \ \ 1\leq i\leq\kappa\leq 2.\end{array}\right.\qquad$$

$$R_{ij\kappa 3}=0,$$

$$\text{II}=Re(\alpha ),$$

$$R_{1212}=-g_{11}{g}_{22}+g_{12}^2,$$

$$\Delta u+1-e^{2u}-\frac{|\alpha {|}^2}{det(g_{\sigma })}{e}^{-2u}=0,$$

$$\Vert \alpha {\Vert }_{\sigma }^2=\frac{|\alpha {|}^2}{det(g_{\sigma })}=\frac{1}{2}\Vert \text{II}{\Vert }_{g_{\sigma }}^2,$$

$$\Delta u+1-e^{2u}-t^2\Vert \alpha {\Vert }_{\sigma }^2{e}^{-2u}=0,$$

$$\underset{S}{\max }|{\tilde{u}}_t|\to +\infty ,$$

$$u_0(z)=\underset{t\nearrow {\tau }_0}{\lim }{u}_t(z)=\underset{t\in (0,{\tau }_0)}{\inf }{u}_t(z),\forall z\in S$$

$$D=\sum _{j=1}^m{\alpha }_j{p}_j,$$

$$|D|=\sum _{j=1}^m{\alpha }_j$$

$$-\Delta u=1-e^{2u}-2\pi \sum _{j=1}^m{\alpha }_j{\delta }_{p_j}.$$

$$\chi (S)+|D|<0,$$

$$t^2\Vert \alpha {\Vert }_{\sigma }^2{e}^{-2{\tilde{u}}_t}\rightharpoonup 4\pi {\delta }_{p_0}$$

$${\tilde{u}}_t\to \tilde{u}\text{ in }{W}^{1,q}(S),1<q<2,\text{ uniformly in }{C}_{loc}^{2,\beta }(S\backslash \{p_0\}),0<\beta <1,$$

$$e^{2{\tilde{u}}_t}\to e^{2\tilde{u}}\text{ in }{L}^s(S),\forall s\ge 1,$$

$$\Delta \tilde{u}+1-e^{2\tilde{u}}-4\pi {\delta }_{p_0}=0.$$

$$\mathcal{J}(w)=\frac{1}{2}{\int }_S|\nabla w{|}^{2}dA-8\pi \log ({\text{fint}}_{S}K(z)e^{w}dA),w\in E,$$

$${\text{fint}}_{S}fdA=\frac{{\int }_{S}fdA}{|S|},$$

$$\underset{E}{\inf }\mathcal{J}>-\infty ,$$

$${\int }_S|{\tilde{u}}_t|dA\to +\infty .$$

$$({\tilde{u}}_{t_n}-{\text{fint}}_S{\tilde{u}}_{t_n})\to {\hat{w}}_0,\text{ strongly in }{H}^1(S),$$

$$t_n^{2}K(z)e^{-2{\tilde{u}}_{t_n}}\to \frac{4\pi K(z)e^{-2{\hat{w}}_0}}{{\int }_{S}K(z)e^{-2{\hat{w}}_0}dA}\text{ uniformly in }{C}^{2,\beta }(S),$$

$$\left\{\begin{array}[]{cl}\Delta\hat{w}_{0}+4\pi\left(\frac{1}{|S|}-\frac{K(z)e^{-2\hat{w}_{0}}}{\int_{S}K(z)e^{-2\hat{w}_{0}}dA}\right)=0\vspace{5pt}\\ \mathcal{J}(-2\hat{w}_{0})=\inf\limits_{E}\mathcal{J};\end{array}\right.\qquad$$

$$t_n^{2}K(z)e^{-2{\tilde{u}}_{t_n}}\rightharpoonup 4\pi {\delta }_{p_0},$$

Full text

0. Introduction: Geometric Settings

Minimal surfaces have long been a fundamental object of intense study in geometry and analysis. In this paper we study minimal immersions of a closed surface in some hyperbolic three-manifolds. Inspired by Uhlenbeck’s approach ([Uhl83]), results on existence and multiplicity of such minimal immersions, as well as their geometrical interpretations, are obtained by analyzing bifurcation properties of solutions to the minimal surface equation. Throughout the paper, we assume $S$ is a closed oriented surface of genus $g\geq 2$. The Teichmüller space of $S$ is denoted by $T_{g}(S)$, and it is the space of conformal structures (or equivalently hyperbolic metrics) on $S$ such that two conformal structures are equivalent if there is between them an orientation-preserving diffeomorphism in the homotopy class of the identity.

When $S$ is immersed in some hyperbolic three-manifold $M$, we denote by $g_{0}$ the induced metric from the immersion and by $\sigma\in T_{g}(S)$ the conformal structure on $S$ induced by $g_{0}$. Furthermore, we denote by $g_{\sigma}$ the unique hyperbolic metric on $(S,\sigma)$, and by $dA$ its area form. Since the metrics $g_{\sigma}$ and $g_{0}$ are conformally equivalent, for a suitable conformal factor $u\in C^{\infty}(S)$, we have

[TABLE]

We denote always by $z=x+iy$ the conformal coordinates on $(S,\sigma)$. So in local conformal coordinates we may write:

[TABLE]

with $v=u_{\sigma}+u$ and $u$ given in (0.1). Now, in such coordinates, the second fundamental form II takes the following quadratic expression:

[TABLE]

with $h_{11}=-h_{22}$ accounting for the fact that $(S,g_{0})$ is a minimal surface in $M$.

The Riemann curvature tensor $R_{ij\kappa\ell}$ and the metric tensor $g=(g_{ij})$ of the hyperbolic three-manifold $(M,g)$ satisfy the following equations:

[TABLE]

Note that by Bianchi identities, only six components of $R_{ij\kappa\ell}$ are independent.

In this respect, we can use normal coordinates $(z,r)$ for the normal bundle $T^{N}(S)$, with conformal coordinates $z\in S$ and $r\in(-a,a)$ for some $a>0$ small. We obtain via the exponential map a local coordinate system on $M$ around $S$, where we have $g_{j3}=\delta_{j3}$, for $j=1,2,3$. In these coordinates, the remaining components $g_{i\kappa}$, $1\leq i\leq\kappa\leq 2$, are just $-R_{i3\kappa 3}$ in view of (0.3), namely,

[TABLE]

The equations in (0.4) can be viewed as a second order system of ODEs for $g_{ik}$ in the variable $r$ (and fixed $z\in S$). So we can uniquely identify $g_{i\kappa}$ (around $S$) by its initial data:

[TABLE]

Such initial data on $S$ are provided by the remaining equations in (0.3). To verify this, we take $\ell=3$ and $j\not=3$ in (0.3) and get

[TABLE]

which expresses the Codazzi equations on $S$, for $1\leq i,j,\kappa\leq 2$. Again only two of those equations are independent, and they ensure exactly that the quadratic differential $\alpha=(h_{11}-ih_{12})dz^{2}$ is holomorphic and

[TABLE]

see [[Hop89, LJ70]]. In other words, $\alpha\in Q(\sigma)$, where we denote by $Q(\sigma)$ the space of holomorphic quadratic differentials on $(S,\sigma)$.

Finally taking $i=\kappa=1$ and $j=\ell=2$ in (0.3), one gets:

[TABLE]

which simply gives the Gauss equation on $S$, and it states that the conformal factor $u(z)$ in (0.1) on $S$ must satisfy:

[TABLE]

with $\alpha$ given in (0.7), and $\Delta$ the Laplacian in the hyperbolic metric $g_{\sigma}$. Indeed, (0.9) simply expresses a consistency condition on $(S,g_{\sigma})$ between the intrinsic curvature of the metric $g_{0}$ and the extrinsic curvature $det_{g_{0}}$II, see [Uhl83] for details.

Note that, by Bianchi identities, once the equations (0.6), (0.8) hold on $S$ then they hold throughout the normal bundle of $S$. Furthermore, these equations on $S$ provide the initial data (0.5) in terms of $(\sigma,\alpha)$, simply by using the solutions of the Codazzi-Gauss equations (0.7), (0.9) into (0.1)-(0.2).

Thus by prescribing $\sigma\in T_{g}(S)$ and $\alpha\in Q(\sigma)$ such that the Codazzi-Gauss equations (0.7), (0.9) are solvable, it is natural to ask whether it is possible to obtain a minimal immersion of $(S,\sigma)$ into some hyperbolic three-manifold with the second fundamental form satisfying (0.7). In short, we shall call a minimal immersion of $S$ with data $(\sigma,\alpha)$ any of such minimal immersion.

A general construction of a minimal immersion with prescribed data satisfying the Codazzi-Gauss equations (called “hyperbolic germs” in [Tau04]) is available in literature, see for instance [Tau04, Jac82]. However, it is not always possible to guarantee that the corresponding hyperbolic three-manifold is complete, unless we are more specific about the induced metric $g_{0}$ or equivalently about the solution of (0.9). Thus, to obtain more satisfactory results of geometrical nature, Uhlenbeck analyzed in [Uhl83] more closely the set of solutions of (0.9).

We recall that a solution $u$ of (0.9) is called stable if the linearized operator of (0.9) at $u$ is nonnegative definite in $H^{1}(S)$, and called strictly stable if the linearized operator of (0.9) at $u$ is positive definite in $H^{1}(S)$. The interest to stable solutions is justified by the fact that they give rise to (local) area minimizing immersions.

By setting:

[TABLE]

the length squared of $\alpha$ with respect to the hyperbolic metric $g_{\sigma}$, Uhlenbeck ([Uhl83]) considered a one-parameter family of Gauss equations:

[TABLE]

for minimal immersions of $S$ with data $(\sigma,t\alpha)$. Using the implicit function theorem, she proved the existence and uniqueness of a smooth solution curve of stable solutions to the equation (0.10):

Theorem 0.1**.**

([Uhl83]) Fixing a conformal structure $\sigma\in T_{g}(S)$, and $\alpha\in Q(\sigma)$, there exists a constant $\tau_{0}>0$, depending only on $(\sigma,\alpha)$, such that the equation (0.10) admits a unique stable solution if and only if $t\in[0,\tau_{0}]$. Furthermore for each $t\in[0,\tau_{0}]$, the stable solution $u_{t}<0$ of (0.10) forms a smooth monotone decreasing curve with respect to $t$. Moreover, $u_{t}$ is strictly stable for $t\in[0,\tau_{0})$ and $u_{t}\nearrow u_{t=0}=0$, as $t\searrow 0$, in $H^{1}(S)$.

From Theorem  0.1, a bifurcation diagram (especially for the lower branch, in absolute value, of the solution curve) can be sketched as below in Figure 1:

In this diagram, Uhlenbeck indicated a first turn of the curve of stable solutions at some $\tau_{0}$, though it is still possible the curve retracts and passes again the value $t=\tau_{0}$ to join other solutions of (0.10) for $t\geq\tau_{0}$. Actually it is our first task here to show that this is never the case.

From the geometrical point of view, by Theorem  0.1 we know that, for $t\in[0,\tau_{0}]$, there exists an area minimizing (stable) immersion of $S$ with data $(\sigma,t\alpha)$ whose induced metric on the surface $(S,\sigma)$ is $g_{0}=e^{2u_{t}}g_{\sigma}$. The second variation of the area functional is explicitly computed in terms of the linearized operator in [Uhl83]. Also it is interesting to note that there exists a $\tau_{1}>0$, such that for each $t\in(0,\tau_{1})$, the surface can be minimally immersed into a so-called almost Fuchsian manifold and this almost Fuchsian manifold contains $(S,\sigma)$ as its unique minimal surface. In particular, as $t\to 0^{+}$ such family of almost Fuchsian hyperbolic three-manifolds converge (in the sense of Gromov-Hausdorff) to a Fuchsian manifold where $S$ is embedded as a totally geodesic area minimizing surface, see [Uhl83] for details.

Further work in [HL12] obtained an additional solution for each of Uhlenbeck’s (strictly) stable solution to the Gauss equation:

Theorem 0.2**.**

([HL12]) Let $S$ be a closed surface and $\sigma\in T_{g}(S)$ be a conformal structure on $S$. If $\alpha\in Q(\sigma)$ is a holomorphic quadratic differential on $(S,\sigma)$, then:

1. i)

for sufficiently large $t$, the Gauss equation (0.10) admits no solutions, i.e., there is no minimal immersion of $S$ with data $(\sigma,t\alpha)$ into any hyperbolic three-manifold; 2. ii)

for each $t\in(0,\tau_{0})$, with $\tau_{0}>0$ given in Theorem  0.1, there exists also an unstable immersion of $S$ with data $(\sigma,t\alpha)$.

These results reveal further details on the solution curve to (0.10) and an improved bifurcation diagram can be sketched as follows:

1. Introduction: Main results

The first purpose of this paper is to complete the above results in Theorems  0.1 and  0.2 as follows. Firstly, we already mentioned, we show that actually the interval $[0,\tau_{0}]$ exhausts the full range of values $t\geq 0$ for which the equation (0.10) is solvable. Namely, the bifurcation curve starting from the trivial solution at $t=0$, cannot admit an “S-shape” (as typical in similar problems), but it turns only at $\tau_{0}$ and until then, furnishes the lower branch (in absolute value) of unique (strictly) stable solutions of (0.10). In fact, equation (0.10) admits a unique (stable but not strictly stable) solution for $t=\tau_{0}$ and no solutions for $t>\tau_{0}$. Furthermore we provide a family of unstable solutions for (0.10) with a specific asymptotic behavior, as $t\to 0^{+}$. A sketch of the bifurcation diagram can be seen as below in Figure 3.

Theorem A**.**

Fixing a conformal structure $\sigma\in T_{g}(S)$, and a holomorphic quadratic differential $\alpha\in Q(\sigma)$, the equation (0.10) admits a solution if and only if $t\in[0,\tau_{0}]$, with $\tau_{0}=\tau_{0}(\sigma,\alpha)>0$ given in Theorem  0.1. Furthermore, the unique stable solution $u_{t}<0$ is the (pointwise) largest solutions of (0.10) for $t\in[0,\tau_{0}]$. Moreover,

1. (i)

$\forall t\in(0,\tau_{0})$, the equation (0.10) admits an unstable solution $\tilde{u}_{t}$ (with $\tilde{u}_{t}<u_{t}<0$ on $S$) such that, as $t\searrow 0$,

[TABLE] 2. (ii)

for $t=\tau_{0}$, the equation (0.10) admits the unique solution $u_{0}$:

[TABLE]

We may say that $(\tau_{0},u_{\tau_{0}})$ is a “bending point” for the bifurcation curve starting at $(t=0,u=0)$, using the terminology introduced in [AR73]. Actually we provide a much more detailed study of the asymptotic behavior of the unstable solution $\tilde{u}_{t}$ in Theorem A, as $t\searrow 0$. Interestingly, its behavior depends on whether the surface $(S,\sigma)$ can be “uniformized” within the class of hyperbolic metrics with conical singularities along a divisor of modulus $2$.

To be more precise, recall that a divisor $D$ on $S$ is a “formal” expression of the type

[TABLE]

with given points $p_{j}\in S$ and values $\alpha_{j}>-1,\ j=1,\cdots,m$; and

[TABLE]

is the modulus of the divisor. Therefore a hyperbolic metric on $S$ with conical singularities along the divisor $D$ in (1.1) is a metric with Gauss curvature $-1$ in $S\backslash\{p_{1},\cdots,p_{m}\}$ and a conical singularity at $p_{j}$ with angle $\theta_{j}=2\pi(1+\alpha_{j})$ for $j=1,\cdots,m$. Thus, the “uniformization” of $(S,\sigma)$ in the class of hyperbolic metrics with conical singularities along the divisor $D$, just means that at least one of such metric is conformal to $g_{\sigma}$. Consequently, the corresponding conformal factor $u$ is given by the unique solution of the following equation:

[TABLE]

By analyzing (1.2), one easily see that the “uniformization” of $S$ in the sense above is possible if and only if

[TABLE]

where $\chi(S)=2(1-g)$ is the Euler characteristic of $S$. In particular, if $|D|=2$, then necessarily the genus $g\geq 3$. Thus our blow-up analysis must distinguish between the cases where $S$ has genus two or higher.

We start with the following:

Theorem B**.**

Let the genus of the surface $S$ satisfy $g\geq 3$, and $\tilde{u}_{t}$ be the unstable solution given by Theorem  A. Then as $t\searrow 0$, we have:

[TABLE]

weakly in the sense of measures, with some point $p_{0}\in S$ such that $\alpha(p_{0})\not=0$. Furthermore:

[TABLE]

and

[TABLE]

with $\tilde{u}$ the unique solution of the following problem on $S$.:

[TABLE]

From the geometrical point of view, Theorem B states that, $\forall t\in(0,\tau_{0})$ there exist (unstable) minimal immersions of $S$ with data $(\sigma,t\alpha)$ which converge (in the sense of Gromov-Hausdorff) as $t\to 0^{+}$ to a “limiting” totally geodesic immersion of $S$ into a three-dimensional hyperbolic cone-manifold of the type introduced by Krasnov-Schlenker [KS07]. More precisely, the given hyperbolic three-manifold contains conical singularities along one line to infinity and the induced metric on $S$ is hyperbolic with a conical singularity along the divisor $D=2p_{0}$, with suitable $p_{0}\in S$ and $\alpha(p_{0})\not=0$. Therefore for our unstable immersions, we observe a quite different limiting behavior from the case of area minimizing (stable) immersions.

If $S$ has genus $g=2$, then the asymptotic behavior of $\tilde{u}_{t}$ is described by a “concentration-compactness” alternative, resolved by the existence (or not) of a minimum for a Moser-Trudinger type functional. To be more precise, let us recall the following:

Definition 1.1**.**

Let $E=\{w\in H^{1}(S)$ with $\int_{S}w(z)dA=0\}$, the Moser-Trudinger functional on $(S,\sigma)$ with weight function $0\leq K\in L^{\infty}(S)$ is given by

[TABLE]

where we have used the standard notation:

[TABLE]

and the area $|S|=4\pi(g-1)$ by the Gauss-Bonnet theorem.

By the Moser-Trudinger inequality (see [Aub98]), the functional $\mathcal{J}$ is bounded from below but not coercive in $E$. In other words, it is well defined

[TABLE]

but the infimum in (1.6) may not be attained.

For the case of genus $g=2$, our main result reads:

Theorem C**.**

If the surface $S$ is of genus $g=2$, and $\tilde{u}_{t}$ is the unstable solution given by Theorem  A, then, as $t\searrow 0$,

[TABLE]

Furthermore, for $K(z)=\|\alpha\|_{\sigma}^{2}$, we have the following alternatives:

1. (i)

(Compactness) either, the Moser-Trudinger functional $\mathcal{J}$ (with $K=\|\alpha\|_{\sigma}^{2}$) attains its infimum in $E$, and along a sequence $t_{n}\to 0$, there holds:

[TABLE]

and,

[TABLE]

with $\hat{w}_{0}$ satisfying on $(S,\sigma)$,

[TABLE] 2. (ii)

(Concentration) or, the functional $\mathcal{J}$ (with $K=\|\alpha\|_{\sigma}^{2}$) does not attain its infimum in $E$, and along a sequence $t_{n}\to 0$, there holds:

[TABLE]

weakly in the sense of measure, for some $p_{0}\in S$ such that $\alpha(p_{0})\not=0$,

[TABLE]

and uniformly in $C_{loc}^{2,\beta}(S\backslash\{p_{0}\}),0<\beta<1$, with $G(\cdot,p)$ the unique Green’s function of the Laplace operator $\Delta$ on the hyperbolic surface $(S,g_{\sigma})$, as defined in (3.1) below.

Hence, from Theorem C we see that, in contrast to higher genera, when $g=2$ the (unstable) minimal immersion of $S$ does not survive the passage to the limit, as $t\to 0^{+}$. Indeed for the induced metric $\tilde{g}_{t}^{0}=e^{2\tilde{u}_{t}}g_{\sigma}$, there holds:

[TABLE]

Nonetheless, when the functional $\mathcal{J}$ in (1.5) with $K=\|\alpha\|_{\sigma}^{2}$ attains its minimum in $E$, then we can still find, as $t\to 0^{+}$, a “limiting” configuration for the “blown-up” surface $(S,\hat{g}_{t}^{0})$ with

[TABLE]

Indeed, along a sequence $t=t_{n}\to 0^{+}$, the blown-up sequence $(S,\hat{g}_{t_{n}}^{0})$ converges (in the sense of Gromov-Hausdorff) to a surface $(S,\hat{g}^{0})$ conformally equivalent to $(S,g_{\sigma})$ with same total negative curvature $-4\pi$. It is interesting to note that $(S,\hat{g}^{0})$ admits a non-positive curvature vanishing exactly as $|\alpha|$.

More generally, for any sequence of unstable solutions $u_{n}$ of (0.10) with $t=t_{n}\to 0^{+}$, as $n\to+\infty$, we shall carry out an analogous blow-up analysis whose details are contained in Theorem D of Section §3. Roughly speaking, to any such sequence, we shall associate a divisor $D$ in $S$ of the following type:

[TABLE]

with suitable $p_{j}\in S$ and $n(p_{j})=0$ if $\alpha(p_{j})\not=0$, while for $p_{j}$ a zero of $\alpha$ (i.e. $\alpha(p_{j})=0$), then $n(p_{j})$ is given by the corresponding multiplicity. Furthermore,

[TABLE]

In case $\chi(S)+|D|<0$, then as before, we find that the sequence of corresponding minimal immersions of $S$ with data $(\sigma,t_{n}\alpha)$ converges (in the sense of Gromov-Hausdorff) to a “limiting” totally geodesic immersion into a three-dimensional hyperbolic cone-manifold which induces on $S$ a hyperbolic metric with conical singularities along the divisor $D$ in (1.10). While in case $\chi(S)+|D|=0$, then no such singular metric exist on $S$ and exactly as above, the given (unstable) minimal immersions do not survive the passage to the limit. Thus, as before, the asymptotic behavior of the blow-up sequence $u_{n}$ is described by a “concentration-compactness” alternative, see Theorem D in section §3 for details.

It is an interesting open problem to see whether unstable solutions of (0.10) can be constructed in such a way that their blow-up behavior matches a prescribed divisor of the type (1.10) and (1.11).

In conclusion, let us make a few remarks.

Remark 1.2**.**

It is well known that any $\alpha\in Q(\sigma)$ admits $4(g-1)$ zeroes, counting multiplicity. Seen from Theorems  B and  C, we have that the blow-up of the unstable solution $\tilde{u}_{t}$, as $t\to 0$, cannot occur around a zero of $\alpha$. A more precise characterization of the blow-up point $p_{0}$ in Theorems  B and  C will be given in the sections §4 and §5.

Remark 1.3**.**

It is interesting to record that for $g=2$, the behavior of the unstable solution $\tilde{u}_{t}$, as $t\to 0^{+}$ depends on whether the Moser-Trudinger functional $\mathcal{J}$ in (1.5) with weight $K=\|\alpha\|_{\sigma}^{2}$ attains in its infimum in $E$. Actually, exactly when $K(z)=\|\alpha\|_{\sigma}^{2}$, the existence of extrema for $\mathcal{J}$ appears to be a delicate open problem. Indeed we shall see in section §5.2 that in such case the functional $\mathcal{J}$ just misses to satisfy the condition provided (for a general weight function $K$) in Theorem 7.2 of [DJLW97]] which is sufficient to ensure the existence of a global minimum for $\mathcal{J}$.

Remark 1.4**.**

From the blow-up analysis in Theorem D where Theorems  B and  C enter as special cases, we shall obtain a compactness result for solutions of (0.10). This will enable us to obtain a minimal immersion of $S$ with prescribed total extrinsic curvature, $\rho=\int_{S}det_{g_{0}}(\text{II})dA(g_{0})\in(0,4\pi(g-1))$ and data $(\sigma,t_{\rho}\alpha)$, with suitable $t_{\rho}\in(0,\tau_{0})$. It would be interesting to investigate the dependence of $t_{\rho}$ with respect to $\rho$.

Remark 1.5**.**

(Added to the proof) The statement of Theorem A and parts of its proof are somewhat similar to a result of Ding-Liu ([DL95]) where they analyzed a problem of prescribing Gaussian curvature on closed surfaces. In fact from the variational point of view, both problems (with a parameter) admit similar structures. More precisely, we have the presence of a (strict) local minimum and a “mountain pass geometry”, for a suitable sharp range of the parameter involved. Typically, in this situation one can claim the existence of a stable and unstable solution (as in Theorem A and in [DL95]) with the stable solution “bifurcating” out of a known (trivial) solution of the problem at a limiting value of the parameter. Much more interesting is the description of the asymptotic behavior of the unstable solution, which reflects the particular nature of the geometrical problem in hands. This is the purpose of our Theorems B, C and D. For the Gauss curvature problem treated in [DL95], this goal has been pursued in [BGS15], and more recently by Struwe in [Str20], who obtained complete results. In fact, Struwe’s result encourages the possibility to also complete our blow-up analysis in Theorem D when blow-up occurs at a zero of the quadratic differential $\alpha$, see Remark 6.3 and 6.4 below. We thank the referee for pointing out these references.

Plan of the rest of the paper:

In §2, we will provide several estimates before we move to prove Theorem  A in section §4. Detailed blow-up analysis is conducted in sections §3 where we obtain Theorem D, and in §5, where we prove Theorems  B and  C. In §6, we extend the program to explore this minimal immersion problem when prescribing the total extrinsic curvature.

Acknowledgements

We wish to thank Biao Wang for his help with the figures. The research of Z.H. was supported by a grant from the Simons Foundation (#359635, Zheng Huang). The research of M.L. was supported by MINECO grant MTM2017-84214-C2-1-P. The research of G.T. was supported by PRIN 2015 Project “Variational methods with application to problems in mathematical physics and geometry” and MIUR excellence project, Department of Mathematics, University of Rome Tor Vargata CUP E83C18000100006.

2. Elementary estimates

Before we proceed, we introduce more convenient notations. We set

[TABLE]

and

[TABLE]

and rewrite the Gauss equation  (0.10) as follows:

[TABLE]

where $v\in H^{1}(S)$, $t\geq 0$, and $K(z)\geq 0$ has finitely many zeroes, given by the zeroes of the prescribed holomorphic quadratic differential $\alpha\in Q(\sigma)$, whose total number is $4g-4$, counting multiplicity.

Definition 2.1**.**

We call a function $v_{t}\in H^{1}(S)$ a solution of problem $(1)_{t}$ for $t\geq 0$, if it solves the equation (2.1).

We collect some basic properties for the solutions of this problem.

Lemma 2.2**.**

If $v$ is a solution of problem $(1)_{t}$, then we have

1. i)

[TABLE]

and in particular,

[TABLE] 2. ii)

$v\geq 0$* and $v\equiv 0$ if and only if $t=0$. Therefore $v(z)>0$ for all $z\in S$ for any $t>0$.* 3. iii)

If we write $v=w+\mathtt{c}$, with $\int_{S}w(z)dA=0$, and $\mathtt{c}=\fint_{S}v(z)dA$, then $\mathtt{c}>0$ and

[TABLE]

Proof.

These properties follow by direct and elementary calculations. More specifically, to obtain (2.2), we integrate (2.1) and apply the Gauss-Bonnet formula. At this point (2.3) is a direct consequence of (2.2) and Schwarz inequality. In order to show (ii), we simply write $v=v^{+}-v^{-}$ where $v^{+}$ and $v^{-}$ are both non-negative. Using $v^{-}$ as a test function we have

[TABLE]

and this is equivalent to

[TABLE]

Since $v^{-}\geq 0$, we find

[TABLE]

But the scalar function $(e^{x}-1)x\geq 0$ whenever $x\geq 0$, we deduce that $v^{-}\equiv 0$ and therefore $v\geq 0$. Furthermore, since $v\in H^{1}(S)$, the righthand side of (2.3) is in $L^{p}$ for any $p\geq 1$. Standard regularity results for elliptic equations imply that $v$ is in fact smooth. Then the conclusion follows from the strong maximum principle. Finally (iii) follows by using $v=w+\mathtt{c}$ in (2.2) and by solving with respect to $\mathtt{c}$. ∎

For later use and according to the choice of the sign in (2.4), we set:

[TABLE]

Lemma 2.3**.**

([HL12]) If $(1)_{t}$ admits a solution, then

[TABLE]

Proof.

Inequality (2.6) easily follows from (2.2) and the Cauchy-Schwarz inequality, as seen in [HL12]. ∎

In particular, this lemma implies that, for $t>t^{\ast}$, there does not exist any minimal immersion of $S$ with data $(\sigma,t\alpha)$.

Lemma 2.4**.**

Let $v=w+\mathtt{c}$ be a solution to problem $(1)_{t}$, where $w$ and $\mathtt{c}$ are as in Lemma  2.2. We have

1. i)

For any $s\in[1,2)$, there exists a constant $C_{s}>0$ such that

[TABLE] 2. ii)

There exists a constant $C$ (independent of $t$) such that:

[TABLE] 3. iii)

For any small $\delta>0$, there exists a constant $C_{\delta}>0$ (independent of $t$) such that,

[TABLE]

for any $t\geq\delta$.

Proof.

Since the righthand side of (2.1) is uniformly bounded in $L^{1}$-norm because of (2.2), independent of $t$, then (2.7) follows from Stampacchia elliptic estimates. In particular, for any $p\geq 1$, there exists a constant $C_{p}>0$ such that $\|w\|_{p}\leq C_{p}$. Denote by $\kappa_{0}=\frac{\|K\|_{\infty}}{\|K\|_{1}}$, then by Jensen’s inequality we find:

[TABLE]

and (2.8) follows.

From (2.5), for $t\geq\delta$, we have:

[TABLE]

and (2.9) follows from (2.8). ∎

In the next section we complete the information of Lemma  2.4 by means of a more accurate blow-up analysis.

3. Blow-up Analysis

In this section we will study in details the general asymptotic behavior of a blow-up sequence of solution for problem $(1)_{t}$. As an application, we will prove Theorems B and C in sections §5 and §6.

Let us denote by $G(q,p)$ the Green’s function (for the hyperbolic Laplace operator) defined as follows:

[TABLE]

Here $\delta_{p}$ is the Dirac measure with pole at the point $p\in S$. It is well-known (see for instance [Aub98]) that

[TABLE]

and

[TABLE]

where $\gamma\in C^{\infty}(S\times S)$ is the regular part of the Green’s function $G$.

3.1. Mean field formulation

Our blow-up analysis about solutions of problem $(1)_{t}$ is based on well-known results concerning blow-up solutions of Liouville type equations in mean field form (see [BM91, LS94, BT02]). Thus, we proceed first to reformulate problem $(1)_{t}$ to a mean field type equation. To this end, we let $v$ be a solution of problem $(1)_{t}$ and set:

[TABLE]

By the conservation identity (2.2), we know that $\rho\in(0,4\pi(g-1))$. As before (in Lemma  2.2), we set $v=w+\mathtt{c}$, where $\int_{S}w(z)dA=0$, and $\mathtt{c}=\fint_{S}v(z)dA$.

Lemma 3.1**.**

If $v=w+\mathtt{c}$ is a solution of the problem $(1)_{t}$ satisfying (3.3) with $\rho\in(0,4\pi(g-1))$, then $w$ satisfies:

[TABLE]

Vice versa, if $w_{\rho}$ solves equation (3.4), with $\rho\in(0,4\pi(g-1))$, then by setting

[TABLE]

and

[TABLE]

we see that $v=w_{\rho}+\mathtt{c}_{\rho}$ is a solution to problem $(1)_{t_{\rho}}$.

Proof.

This follows by direct and simple calculations. ∎

3.2. General blow-up

Recall that $K(z)=\frac{|\alpha(z)|^{2}}{det(g_{\sigma})}$. Denote by $\{q_{1},\cdots,q_{N}\}$ the (finite) set of distinct zeroes of $\alpha$, i.e.,

[TABLE]

and let $n_{i}$ be the multipilcity of $q_{i}$. It is well known that, $\sum\limits_{i=1}^{N}n_{i}=4(g-1)$.

Our main result in this section is the following theorem, which in particular indicates that blow-up occurs only when $t\to 0$.

Theorem D**.**

Let $v_{n}$ be a solution of the problem $(1)_{t_{n}}$ such that

[TABLE]

then, as $n\to\infty$,

[TABLE]

and

[TABLE]

where $g\geq 2$ is the genus of $S$. Furthermore,

1. (i)

if $1\leq m<g-1$, then (along a subsequence),

1. (a)

there exist $\{p_{1},\cdots,p_{s}\}\subset S$ (called blow-up points), and sequences $\{p_{j,n}\}\subset S$ such that $p_{j,n}\to p_{j}$ and $v_{n}(p_{j,n})\to+\infty$ as $n\to+\infty,\ j=1,\cdots,s$. Moreover,

[TABLE]

weakly in the sense of measure, with

[TABLE]

and  $m=\sum\limits_{j=1}^{s}(1+n(p_{j}))$. 2. (b)

$v_{n}\rightharpoonup v_{0}$* weakly in $W^{1,q}(S)$, for some $v_{0}$ and $1<q<2$, and uniformly in $C_{loc}^{2,\beta}(S\backslash\{p_{1},\cdots,p_{s}\}$ for some $\beta\in(0,1)$. We also have*

[TABLE]

strongly in $L^{p}(S)$ for all $p\geq 1$, and $v_{0}$ is the unique solution of the following equation on $S$:

[TABLE] 2. (ii)

If $m=g-1$, then by setting $v_{n}=w_{n}+\mathtt{c}_{n}$, where $\int_{S}w_{n}(z)dA=0$, and $\mathtt{c}_{n}=\fint_{S}v_{n}(z)dA$, we have (along a subsequence):

[TABLE]

Furthermore we have the following alternatives:

1. (a)

(Compactness) either,

[TABLE]

and

[TABLE]

with $w_{0}$ satisfying the following equation on $S$:

[TABLE] 2. (b)

(Concentration) or, for suitable points $\{p_{1},\cdots,p_{s}\}\subset S$ (blow-up points), we have sequences $\{p_{i,n}\}\subset S$: $p_{i,n}\to p_{i}$ such that:

[TABLE]

and:

[TABLE]

weakly in the sense of measure, where $n(p_{i})$ is defined in (3.8) with $\sum_{i=1}^{s}(1+n(p_{i}))=g-1$, and $w_{0}(z)$ satisfying:

[TABLE]

with $G(z,p)$ the unique Green’s function in (3.1). The convergence is uniform in $C_{loc}^{2,\beta}(S\backslash\{p_{1},\cdots,p_{s}\})$.

Proof.

As in the statement, we write $v_{n}=w_{n}+\mathtt{c}_{n}$, where $\int_{S}w_{n}(z)dA=0$, and $\mathtt{c}_{n}=\fint_{S}v_{n}(z)dA$. By Lemma  2.4, we can always assume that, along a subsequence,

[TABLE]

Let

[TABLE]

So we can write $\int_{S}e^{-v_{n}}dA=4\pi(g-1)-\rho_{n}$ (recall (2.2)).

Denote by $\zeta_{n}$ the unique solution to the problem:

[TABLE]

Recall that $v_{n}>0$ in $S$, and

[TABLE]

Therefore, we know that the righthand side of (3.14) is uniformly bounded in $L^{\infty}(S)$. Thus, by elliptic estimates, we derive that $\zeta_{n}$ is uniformly bounded in $C^{2,\beta}(S)$-norm, with $\beta\in(0,1)$. So, along a subsequence, we can assume that,

[TABLE]

We define, somewhat abusing our notation to use $z_{n}$ as follows,

[TABLE]

which satisfies the following mean field type equation:

[TABLE]

with

[TABLE]

At this point, we recall the following well-known “concentration-compactness” result of [BM91, LS94, BT02] which we state in a form suitable for our situation:

Theorem 3.2**.**

([BM91, LS94, BT02]) Assume (3.17) with $\rho_{n}\in(0,4\pi(g-1))$ and (3.18) with $K(z)=\frac{|\alpha|^{2}}{det(g_{\sigma})}$, then, along a subsequence, as $n\to+\infty$:

[TABLE]

and the following alternative holds:

1. (1)

either*, $\max_{S}z_{n}\leq C$, and along a subsequence, as $n\to+\infty$:*

[TABLE]

[TABLE]

and in any other relevant norm, with $z_{0}$ satisfying:

[TABLE]

with $h=Ke^{\zeta_{0}}$ (see (3.15)). 2. (2)

or*, there exist (blow-up) points $\{p_{1},\cdots,p_{s}\}\subset S$, and sequences $\{p_{j,n}\}\subset S$: $p_{j,n}\to p_{j}$, such that $z_{n}(p_{j,n})\to+\infty$ as $n\to+\infty$, such that,*

[TABLE]

weakly in the sense of measure, with $n(p_{j})$ defined in (3.8). In particular,

[TABLE]

[TABLE]

and

[TABLE]

Proof.

See Theorem 5.7.65 in [Tar08]. ∎

Back to the proof of Theorem D. We apply Theorem  3.2 to $z_{n}$ in (3.16), and along a subsequence, we have:

[TABLE]

Recall that by assumption: $\max\limits_{S}v_{n}\to\infty$, and so in case alternative (1) in Theorem  3.2 holds, in view of (3.15) and (3.16), we see that necessarily

[TABLE]

Therefore, by dominated convergence, we derive:

[TABLE]

So $\rho_{n}\to 4\pi(g-1)$, and by (3.13) we deduce that (3.6) must hold with $m=g-1$ in this case.

This covers the compactness part in the statement (ii). Furthermore, by part (iii) of Lemma  2.4, we see also that $t_{n}\to 0$, as $n\to\infty$, and so (3.5) holds.

Next we assume that alternative (2) in Theorem  3.2 holds. Then by virtue of (3.13), (3.20), and (3.21), we check that (3.6) holds with $m:=\sum\limits_{j=1}^{s}(1+n(p_{j}))$. We consider first the case where $1\leq m<g-1$. As a consequence,

[TABLE]

and so necessarily $\mathtt{c}_{n}=\fint_{S}v_{n}$ must be uniformly bounded. Thus along a subsequence, we have: $v_{n}\rightharpoonup v_{0}$ weakly in $W^{1,q}(S)$, $1<q<2$, and also uniformly in $C^{2,\beta}(S\backslash\{p_{1},\cdots,p_{s}\})$ with $0<\beta<1$, for some $v_{0}$. Furthermore, by dominated convergence, we see that

[TABLE]

As a consequence $v_{0}$ satisfies (3.10). In other words, we have verified that part (i) holds in this case.

Since,

[TABLE]

uniformly on compact sets of $S\backslash\{p_{1},\cdots,p_{s}\}$, we may conclude that (3.5) must hold as well.

Finally, when we have alternative $(2)$ with $m=g-1$, then necessarily

[TABLE]

As a consequence, $\mathtt{c}_{n}\to+\infty$, as $n\to\infty$ and this implies as above, that $t_{n}\to 0$, by part (c) of Lemma  2.4. Thus we have verified (3.5), (3.11), and (3.6) with $m=g-1$. At this point, alternative (2) of Theorem 3.2 in this situation gives exactly the (concentration) part (b) of (ii).

Finally, since $w_{0}$ in (3.11) satisfies:

[TABLE]

we see that (3.12) holds, and the proof is complete. ∎

Concerning the location of the (possible) blow-up points $\{p_{1},\cdots,p_{s}\}$ of $v_{n}$, we can use well-known results ([OS05]) which apply to the blow-up points of the sequence $z_{n}$ in (3.17). Thus, according to Theorem 2.2 of ([OS05]), we conclude that, if $p_{j}$ is a blow-up point with $\alpha(p_{j})\not=0$, then in conformal coordinates around $p_{j}$ there holds:

[TABLE]

with $i\in\{1,\cdots,s\}$, and

[TABLE]

with $\zeta_{0}$ in (3.15) and $K=\|\alpha\|_{\sigma}^{2}$.

Notice in particular that when $m=g-1$, then the function $\zeta_{0}\equiv 0$, and (3.24) provides a well-known necessary condition for blow-up at $\{p_{1},\cdots,p_{s}\}$. Indeed, in case of non-degeneracy, (3.24) turns out to be also a sufficient condition for the construction of blow-up solutions at $\{p_{1},\cdots,p_{s}\}$ for mean field equations on surfaces, see for instance [CL03].

On the contrary, when $1\leq m<g-1$, the condition (3.24) is more involved since the function $\zeta_{0}$ is nonzero and satisfies the following equation:

[TABLE]

So $\zeta_{0}$ itself depends on the blow-up points $\{p_{1},\cdots,p_{s}\}$. Therefore it would be interesting to see whether one can find a (nondegenerate) set of points satisfying (3.24), (3.25) and (3.26) which turns out to be the blow-up set of a sequence of bubbling solutions for $(1)_{t}$, along a sequence of $t$’s going to zero.

As a consequence of Theorem D, we know that blow-up can only occur as $t\to 0$. Therefore, we can complete the (uniform) estimates given in Lemma  2.4 as follows:

Corollary 3.3**.**

For any $\delta>0$, there exists a constant $C_{\delta}>0$ such that any solution $v$ of the problem $(1)_{t}$ with $t\geq\delta$ satisfies:

[TABLE]

Actually, by means of elliptic estimates we know that the $L^{\infty}(S)-$norm above can be replaced by any other stronger norm. Theorem D can be better interpreted in terms of the mean field formulation of problem $(1)_{t}$, and gives the following “compactness” result:

Corollary 3.4**.**

Let $w_{n}$ be a sequence of solutions for (3.4) with $\rho=\rho_{n}$, and $\rho_{n}\to\rho_{0}\in(0,4\pi(g-1)]\backslash\{4\pi m,1\leq m\leq g-1\}$. Then along a subsequence, $w_{n}\to w_{0}$ in $H^{1}(S)$ (and any other relevant norm), with $w_{0}$ a solution of (3.4) with $\rho=\rho_{0}$.

Corollary 3.5**.**

For every compact set $\mathcal{A}\subset(0,4\pi(g-1)]\backslash\{4\pi m,1\leq m\leq g-1\}$, the set of solutions of (3.4) with $\rho\in\mathcal{A}$ is uniformly bounded in $C^{2,\beta}(S),\ 0<\beta<1$.

4. Proof of theorem  A

In this section, we will prove parts of Theorem  A, in various steps. In this way, we obtain a detailed description of the lower branch of the bifurcation solution curve $\mathcal{C}$. We shall take advantage of the variational formulation of problem $(1)_{t}$. Indeed, it is easy to verify that (weak) solutions of problem $(1)_{t}$ correspond to the critical points of the following functional:

[TABLE]

4.1. First bending point

We define the following two sets:

[TABLE]

and

[TABLE]

Clearly $\Lambda_{s}\subseteq\Lambda\subset[0,t^{\ast}]$, with $t^{\ast}$ given in (2.6). Furthermore, since the problem $(1)_{t=0}$ only admits the trivial solution $v=0$, which is strictly stable, we see that, $\Lambda_{s}$ is nonempty and,

[TABLE]

We aim to show that $\Lambda=\Lambda_{s}$ and $t_{0}=\tau_{0}$.

To this purpose, we observe firstly that, by the estimates in Corollary  3.3 and a limiting argument, we know that problem $(1)_{\tau_{0}}$ admits a stable solution $v_{0}$ which is also degenerate. According to the language of Crandell and Rabinowitz ([CR80]), $(v_{0},\tau_{0})$ defines a “bending point” for the curve of solutions of problem $(1)_{t}$, given by the zero set of the map

[TABLE]

with $0<\beta<1$.

To establish Theorem  0.1, Uhlenbeck ([Uhl83]) showed that actually $\tau_{0}$ is the only value for which the problem $(1)_{t}$ admits a degenerate stable solution.

Proposition 4.1**.**

The problem $(1)_{t}$ admits a degenerate stable solution only at $t=\tau_{0}$. Moreover, for any $t\in[0,\tau_{0}]$ problem $(1)_{t}$ admits a unique stable solution which forms a smooth monotone increasing curve (with respect to $t$), and it is strictly stable for $t\in[0,\tau_{0})$.

Proof.

Let $v_{0}$ be the degenerate stable solution for $(1)_{\tau_{0}}$. We know that $(v_{0},\tau_{0})$ must correspond to a “bending point” for the set: $F(v,t)=0$, around $(v_{0},\tau_{0})$ (see [CR80]).In other words, for $\epsilon>0$ small, there exists a smooth curve $(v(s),t(s))$ satisfying $F(v(s),t(s))=0$ for all $s\in(-\epsilon,\epsilon)$, such that: $v(0)=v_{0},\ t(0)=\tau_{0}$, $\dot{t}(0)=0$, and $\dot{v}>0$ (i.e. $v(s)$ is increasing), where we have used the “dot” to denote derivatives with respect to $s$.

Uhlenbeck showed further that $\ddot{t}(0)<0$ (see [Uhl83]), so that:

[TABLE]

and in particular, $\dot{t}(s)>0$ for $s\in(-\epsilon,0)$. This implies that $v(s)$ is strictly stable for every $s\in(-\epsilon,0)$.

Note that the same local description would hold for any other (possible) degenerate stable solution for which the corresponding of (4.3) would hold. This shows that if we continue the lower branch $(v(s),t(s)),\ s\in(-\epsilon,0)$ with the Implicit Function Theorem, we see that it cannot join with another degenerate stable solution at lower value of $t$. Instead, the lower branch can be continued until it joins the trivial solution at $t=0$. Thus, we can conclude that there exists a smooth, increasing curve of strictly stable solutions of problem $(1)_{t},\ t\in[0,\tau_{0})$, which joins the trivial solution $v=0$ at $t=0$ with the degenerate stable solution $v_{0}$ at $t=\tau_{0}$. This also shows that for any $t\in[0,\tau_{0}]$, problem $(1)_{t}$ cannot admit any other stable solution. As otherwise we could argue as before to join such a different solution to the trivial solution along another smooth curve of solutions, a contradiction to the non-denegeracy of the trivial solution $v\equiv 0$ at $t=0$. This concludes the proof. ∎

Proposition  4.1 shows in particular that $\Lambda_{s}=[0,\tau_{0}]$ and it also furnishes a proof to Theorem  0.1.

In the next subsection, we shall prove, with the help of the sub/super solution method in variational guise (see [Str00]), that problem $(1)_{t}$ admits a stable solution for any $t\in[0,t_{0}]$ where $t_{0}=\sup\{\Lambda\}$. Consequently $\Lambda=\Lambda_{s}=[0.\tau_{0}]$.

4.2. Stable solutions

We now prove the following theorem on stable solutions:

Theorem 4.2**.**

Let $t_{0}=\sup\Lambda$ ($\Lambda$ in (4.2)). For any $t\in[0,t_{0}]$, problem $(1)_{t}$ admits a stable solution $v_{1,t}$ which is strictly increasing with respect to $t$, and coincides with the smallest of the solutions (and supersolutions) of Problem $(1)_{t}$. Furthermore, for $t=t_{0}$,

[TABLE]

is the unique solution for problem $(1)_{t_{0}}$. In particular: $t_{0}=\tau_{0}$ and $\Lambda_{s}=\Lambda=[0,\tau_{0}]$.

Proof.

Since for $t=0$, $v\equiv 0$ is the desired stable solution, we let $t\in(0,t_{0})$ be fixed. By (4.4), we can find some $t_{1}\in\Lambda$ such that, $0<t<t_{1}$. We denote by $v_{1}>0$ a solution for problem $(1)_{t_{1}}$, and observe that it defines a strict super-solution for problem $(1)_{t}$. Indeed, we have

[TABLE]

for any $\phi\in H^{1}(S)$ with $\phi\geq 0$ a.e. in $S$, and $\phi\not\equiv 0$.

While $v_{0}\equiv 0$ is an obvious strict sub-solution for problem $(1)_{t}$. We set

[TABLE]

It is routine to verify that $\mathcal{Z}$ is a non-empty, convex and closed subset of $H^{1}(S)$. In addition, the functional $\mathcal{I}_{t}$ is bounded from below and coercive on $\mathcal{Z}$. Consequently the functional $\mathcal{I}_{t}$ attains its minimum value at a point $v_{1,t}$ in $\mathcal{Z}$, i.e.,

[TABLE]

By the strict sub/super solution property of $v_{0}\equiv 0$ and $v_{1}$ respectively, we have

[TABLE]

and therefore $v_{1,t}\not\equiv v_{1}$ and $v_{t}\not\equiv 0$. Following [Str00], we show next that $v_{1,t}$ is a critical point for $\mathcal{I}_{t}$, therefore a solution to $(1)_{t}$. We first let $\phi\in C^{\infty}(S)$ with $\phi\geq 0$, and for $\epsilon>0$ sufficiently small, we define

[TABLE]

where

[TABLE]

and

[TABLE]

Therefore we have $v_{\epsilon}\in\mathcal{Z}$. By virtue of (4.6) and (4.7), we have:

[TABLE]

We define a set

[TABLE]

We observe that, $|\Omega_{\epsilon}|$ the measure of $\Omega_{\epsilon}$ goes to zero as $\epsilon\to 0$. Moreover,

[TABLE]

Similar calculations show that

[TABLE]

Applying (4.8), we find:

[TABLE]

We obtain the reverse inequality by replacing $\phi$ with $-\phi$. Since $C^{\infty}(S)$ is dense in $H^{1}(S)$, by a density argument we have now proved:

[TABLE]

This implies that $v_{1,t}$ is a solution to problem $(1)_{t}$. Note that $v_{1,t}\not\equiv 0$ and $v_{1,t}\not\equiv v_{1}$, so by the maximum principle, we have:

[TABLE]

This ensures that $v_{1,t}$ is a local minimum for the functional $\mathcal{I}_{t}$ in $C^{1}(S)$-norm.

We are going to show that $v_{1,t}$ is actually a local minimum for $\mathcal{I}_{t}$ in $H^{1}(S)$-norm, and hence a stable solution for problem $(1)_{t}$. To this purpose, we argue by contradiction. Suppose there exists $v_{n}\in H^{1}(S)$ such that $\mathcal{I}_{t}(v_{n})<\mathcal{I}_{t}(v_{1,t})$, and $v_{n}\to v_{1,t}$ in $H^{1}(S)$. Letting

[TABLE]

we may assume, without loss of generality, that

[TABLE]

This enables us to apply the Lagrange multiplier method, and for suitable $\lambda_{n}\in\mathbb{R}$, we find that $v_{n}$ satisfies

[TABLE]

We set

[TABLE]

Then we see that $\|\eta_{n}\|_{H^{1}}\leq 1$. So along a subsequence, $\eta_{n}$ converges weakly in $H^{1}(S)$ to some function $\eta\in H^{1}(S)$, as $n\to+\infty$. Moreover, for any $p\geq 1$ we also have

[TABLE]

By recalling that $\mathcal{I}^{\prime}_{t}(v_{1,t})=0$, we compute:

[TABLE]

Since $v_{n}\to v_{1,t}$ in $H^{1}(S)$, we know that, for any $p\geq 1$,

[TABLE]

Therefore by view of (4.13) we may conclude that

[TABLE]

and

[TABLE]

We claim that $\eta\not=0$. To see this, by contradiction we assume $\eta=0$, then

[TABLE]

as $n\to\infty$. From (4.13), (4.14) and (4.15), we find:

[TABLE]

which is impossible. Therefore $\eta\not=0$.

Using (2.1) for $v_{1,t}$ and (4.10), we see that:

[TABLE]

As above we find, as $n\to\infty$,

[TABLE]

and

[TABLE]

Now we have:

[TABLE]

So by (4.17) and (4.20), we can use elliptic regularity theory to conclude that $v_{n}\in C^{1}(S)$. Furthermore, the righthand side of (4.17) converges to zero in $L^{p}(S)$, for $p>1$, as $n\to+\infty$. Consequently, by (4.20), we can use again elliptic estimates to show that $(v_{n}-v_{1,t})\to 0$ in $C^{1}$-norm. This is a contradiction to the fact that $v_{1,t}$ is a local minimizer for the functional $\mathcal{I}_{t}$ in $C^{1}$-norm. Therefore $v_{1,t}\in H^{1}(S)$ is a local minimum of $\mathcal{I}_{t}$ and hence a stable solution for the problem $(1)_{t}$ with $t\in(0,t_{0})$.

So far we have shown that $\forall t\in[0,t_{0})$, problem $(1)_{t}$ admits a stable solution $v_{1,t}$ (and infinitely many supersolutions). As a consequence,

[TABLE]

and by Proposition  4.1 we know also that problem $(1)_{\tau_{0}}$ admits a unique stable (degenerate) solution $v_{0}$.

For $t\in(0,\tau_{0})$, to show that $v_{1,t}$ is the smallest among all solutions (and supersolutions) of problem $(1)_{t}$, we define for $z\in S$,

[TABLE]

Clearly, $v_{t}(z)$ defines a supersolution of problem $(1)_{t}$, in the sense that the following holds:

[TABLE]

for any $\phi\in H^{1}(S)$ with $\phi\geq 0$. Since $t>0$, we have $v_{t}\not\equiv 0$. Moreover, (4.22) can never hold with a strict sign, as otherwise we would be in position to apply the sub/super solution method as above, and obtain a solution of problem $(1)_{t}$ which is smaller than $v_{t}$, in contradiction with (4.21). Hence $v_{t}$ is a solution of problem $(1)_{t}$ which, by definition, is the smallest solution of $(1)_{t}$ and strictly increasing with respect to $t\in[0,\tau_{0})$.

We claim:

[TABLE]

To establish this claim, it suffices to show that $v_{t}$ is stable, so that (4.23) follows by the uniqueness of stable solutions in Proposition  4.1.

To this purpose, for $t\in(0,\tau_{0})$, we use $v_{t}$ as a supersolution to problem $(1)_{s}$, for $0<s<t$. As above, for $s\in(0,t)$, we obtain a stable solution $\bar{v}_{s}$ of problem $(1)_{s}$ satisfying $0<\bar{v}_{s}<v_{t}$ in $S$. By taking a sequence $s_{n}\nearrow t$, then by dominated convergence and elliptic estimates, we see that $\bar{v}_{s_{n}}\to\bar{v}$ in $H^{1}(S)$, with $\bar{v}$ a stable solution of problem $(1)_{t}$ and $0<\bar{v}\leq v_{t}$. Since $v_{t}$ is the smallest solution to $(1)_{t}$, we conclude that $\bar{v}\equiv v_{t}$, and so $v_{t}$ is stable and (4.23) is established.

From (4.23), it also follows that, $\forall t\in[0,\tau_{0})$, $v_{1,t}<v_{0}$, with $v_{0}$ the unique stable solution of problem $(1)_{\tau_{0}}$. So by the monotonicity property of $v_{1,t}$ in $t$, we find:

[TABLE]

where again by dominated convergence and elliptic estimates, the convergence actually occurs uniformly in $C^{2,\beta}(S),0<\beta<1$. Clearly, $v_{0}$ must define the smallest solution of problem $(1)_{\tau_{0}}$, i.e., $v_{0}=v_{1,\tau_{0}}$. In fact we show that actually $v_{0}$ is the only solution of problem $(1)_{\tau_{0}}$.

To this purpose we argue by contradiction and assume there is another solution $v^{\prime}$ for the problem $(1)_{\tau_{0}}$. By construction, $v_{0}$ is the smallest solution at $t=\tau_{0}$, so necessarily $v^{\prime}>v_{0}$ on $S$. As seen in Proposition  4.1, around $(v_{0},\tau_{0})$, we find a solution curve $(v(s),t(s))$ such that for $s\in(0,\epsilon)$ and $\epsilon>0$ sufficiently small, we obtain a solution $v(s)$ for the problem $(1)_{t(s)}$ such that $t(s)<\tau_{0}$ and $v_{0}<v(s)<v^{\prime}$. Thus for $t\in(t(s),\tau_{0})$ we find $v(s)$ as subsolution and $v^{\prime}$ as supersolution for problem $(1)_{t}$. So we can use the sub/super solution method again, and for $t\in(t(s),\tau_{0})$, we obtain a stable solution for the problem $(1)_{t}$ which will be greater than $v_{0}$, and therefore greater than the smallest solution $v_{1,t}$. This is impossible, since the smallest solution is also the only (strictly) stable solution of problem $(1)_{t}$. ∎

4.3. Compactness for the functional $\mathcal{I}_{t}$

We will complete the proof of Theorem  A in this subsection, namely we will prove the existence of an additional solution for each $t\in(0,t_{0})$. This will extend a multiplicity result in ([HL12]).

By combining Proposition  4.1 and Theorem  4.2, we have now established that, $\forall\ t\in(0,\tau_{0})$, $v_{1,t}$ is a strict local minimum for $\mathcal{I}_{t}(v)$ in $H^{1}(S)$. Furthermore, one checks that, for every $t\in(0,\tau_{0})$:

[TABLE]

as $C\to+\infty$. In other words, for $t\in(0,\tau_{0})$, the functional $\mathcal{I}_{t}$ admits a “mountain-pass” structure ([AR73]). Next we establish the following Palais-Smale (compactness) condition:

Theorem 4.3**.**

Suppose a sequence $\{v_{n}\}\in H^{1}(S)$ satisfies that, $\mathcal{I}_{t}(v_{n})\to c$ and $\mathcal{I}^{\prime}_{t}(v_{n})\to 0$ as $n\to\infty$, then $\{v_{n}\}$ admits a convergent subsequence. In particular, $c$ is a critical value for the functional $\mathcal{I}_{t}$.

Proof.

We show first that $v_{n}$ is uniformly bounded in $H^{1}$-norm. As in Lemma  2.2, we write $v_{n}=w_{n}+\mathtt{c}_{n}$, with $\int_{S}w_{n}(z)dA=0$, and $\mathtt{c}_{n}=\fint_{S}v_{n}(z)dA$. Then, we have

[TABLE]

By assumption, $\langle\mathcal{I}^{\prime}_{t}(v_{n}),1\rangle=o(1)$ as $n\to\infty$. Applying Jensen’s inequality, we have, as $n\to\infty$:

[TABLE]

Therefore for some suitable constant $C_{0}>0$, we find $\mathtt{c}_{n}\geq-C_{0}$. Now from (4.1) and (4.24), we obtain, as $n\to\infty$:

[TABLE]

By assumption, $\mathcal{I}(v_{n})$ is uniformly bounded, so from (4.25) we also see that $\mathtt{c}_{n}$ is bounded from above, and that $\|\nabla w_{n}\|_{2}$ is uniformly bounded.

In conclusion we have $\|v_{n}\|_{H^{1}}\leq C$ for some suitable $C>0$. Therefore along a subsequence, $v_{n}$ converges to some $v\in H^{1}(S)$ weakly. The convergence is strong in $L^{p}(S)$ for $p\geq 1$. In particular we have $\mathtt{c}_{n}\to\fint_{S}vdA$ and $\mathcal{I}^{\prime}(v)=0$. By the Moser-Trudinger inequality, we also have:

[TABLE]

In particular, for $p=2$, and as $n\to\infty$,

[TABLE]

In other words, we have

[TABLE]

as $n\to\infty$. This completes the proof. ∎

We can now apply the mountain pass construction of Ambrosetti-Rabinowitz ([AR73]) to obtain a second (unstable) mountain pass solution $v_{2,t}>v_{1,t}$ for all $t\in(0,t_{0})$, satisfying:

[TABLE]

with the path space

[TABLE]

Clearly $\mathcal{P}_{t}$ is not empty, since for $A>0$ sufficiently large we easily check that $\Gamma(s)=v_{1,t}+sA,\ s\in[0,1]$ lies in $\mathcal{P}_{t}$.

Finally we show that the unstable solution $v_{2,t}$ will not stay bounded as $t\to 0$:

Proposition 4.4**.**

For $t\in(0,\tau_{0})$, let $v_{2,t}$ be the mountain pass solution obtained above. Then:

[TABLE]

Proof.

We argue by contradiction. Suppose that, along a sequence $t_{n}\to 0$, we have

[TABLE]

for suitable constant $C>0$. Then by elliptic estimates (along a subsequence), we find that $\{v_{2,t_{n}}\}$ converges strongly in $C^{2,\beta}(S)$ norm to $v\equiv 0$, the unique solution of problem $(1)_{t=0}$. But this is impossible, since for $t>0$ small, the stable solution $v_{1,t}$ is the only solution of $(1)_{t}$ contained in a small ball centered at the origin . ∎

Clearly, by a similar argument, we see that any family of unstable solutions of (0.10) admits the same blow-up behavior in (4.28), as $t\to 0^{+}$.

5. Blow-up analysis: applications to mountain pass solutions

In this section, we apply the general blow-up analysis of §3 to the mountain pass solution $v_{2,t}$ of problem $(1)_{t}$ obtained in Theorem A. The asymptotic behavior of $v_{2,t}$ differs when the surface has genus two or higher.

By Proposition  4.4 and Theorem D, we know that:

[TABLE]

for suitable $m\in\mathbb{N}$ satisfying $1\leq m\leq g-1$.

Our first goal is to prove that actually, $m=1$. We start with the case where the genus of the surface $S$ is at least three.

5.1. Blow-up analysis when $g\geq 3$

Theorem 5.1**.**

Let the genus $g\geq 3$. Then for $K=\|\alpha\|_{\sigma}^{2}$ we have:

1. (i)

[TABLE] 2. (ii)

As $t\to 0$,

[TABLE]

with some suitable $p_{0}\in S$ such that $K(p_{0})\not=0$ (i.e. $\alpha(p_{0})\not=0$). 3. (iii)

[TABLE]

[TABLE]

and,

[TABLE]

Moreover, $v_{0}$ is the unique solution to the following equation on $S$:

[TABLE]

In order to establish Theorem  5.1, we establish first the following estimates:

Lemma 5.2**.**

If the genus $g\geq 3$, then for a suitable constant $C>0$, we have:

[TABLE]

and

[TABLE]

Proof.

By virtue of Corollary  3.3, clearly it suffices to prove (5.3) and (5.4) as $t\searrow 0$. We start by showing:

[TABLE]

with a suitable constant $C>0$ (independent of $t$).

To this purpose we use sharp estimates obtained in [DJLW97] in order to establish the existence of minimizers for the Moser-Trudinger functional $\mathcal{J}$ in (1.5). We fix $p\in S$ with $K(p)\not=0$. As in [DJLW97], we use normal (polar) coordinates at $p$, centered at the origin, so that for $r=dist(q,p)$, we have:

[TABLE]

with $A(p)=8\pi\gamma(p,p)$, suitable constants $b_{1}$ and $b_{2}$ depending on the hyperbolic metric $g_{\sigma}$, and $\beta(r,\theta)=o(r)$ as $r\to 0$. Recall that $G(p,q)$ is the Green’s function in (3.2), and $\gamma(p,q)$ its regular part.

We let $\eta$ be a standard cut-off function such that:

[TABLE]

where $a_{t}>0$ is chosen in such a way that $a_{t}\to 0$ and $\alpha_{t}=\frac{a_{t}}{t}\to\infty$, as $t\to 0^{+}$.

Now we let,

[TABLE]

For $\varphi_{t}$, we can use well-known estimates. For example from the much sharper estimates derived in [DJLW97] that we apply with $\epsilon=t^{2}$, $\phi_{\epsilon}=\varphi_{t}+\log t^{2}$, and $\alpha=\alpha_{t}$, we obtain that, as $t\to 0$,

[TABLE]

[TABLE]

and

[TABLE]

Next we construct a suitable path in $\mathcal{P}_{t}$ (defined in (4.27)) as follows:

[TABLE]

with $\tilde{c}_{t}\gg 1$ sufficiently large to ensure that

[TABLE]

Clearly $\Gamma_{t}\in\mathcal{P}_{t}$. Furthermore, by virtue of the above estimates (5.6), (5.7), (5.8), for $t>0$ sufficiently small, we have:

[TABLE]

for some suitable $C>0$, independent of $t$. In view of (4.26), this proves (5.5).

To obtain the reverse inequality, we decompose:

[TABLE]

We use the Moser-Trudinger inequality (see for instance [Aub82]) to estimate:

[TABLE]

By (5.1), it is necessary that:

[TABLE]

and so from (5.9) we find that,

[TABLE]

for some suitable constant $C_{0}>0$.

As a consequence, we find:

[TABLE]

We are assuming $g\geq 3$, and also we know that: $\mathtt{c}_{t}>0$ and $t^{2}\int_{S}Ke^{v_{2,t}}dA\in(0,4\pi(g-1))$. Thus, from (5.12) we easily derive (5.3) and (5.4). ∎

Now we will prove Theorem  5.1:

Proof.

(of Theorem  5.1) Recall that we have set, $v_{2,t}=w_{t}+\mathtt{c}_{t}$, and from (5.3) and (5.4), it follows that, as $t\searrow 0$,

[TABLE]

While by the first estimate in (5.9) and (5.10), we also have:

[TABLE]

As a consequence of (5.13), (5.14) and the Moser-Trudinger inequality, as $t\searrow 0$, we find:

[TABLE]

Moreover, by (5.3) and Proposition  4.4, it follows that,

[TABLE]

By the improved Moser-Trudinger inequality of Chen-Li [CL91] (see Lemma 6.2.7 in [Tar08] and also Malchiodi-Ruiz [MR11]), and in view of (5.15) and (5.16), we have that, there exists a unique point $p_{0}\in S$, such that, $\forall r>0$ sufficiently small, the following holds as $t\searrow 0$:

[TABLE]

[TABLE]

and

[TABLE]

for a suitable constant $C_{r}>0$. In other words, $p_{0}$ is the unique blow-up point for $w_{t_{n}}$, along any sequence $t_{n}\searrow 0$, see [BM91]. That is, if $p_{n}\in S$ satisfies:

[TABLE]

then:

[TABLE]

We shall show that,

[TABLE]

and since $K=\|\alpha\|_{\sigma}^{2}$, $p_{0}$ cannot be a zero for $\alpha\in Q(\sigma)$.

In order to see this, we use (5.3) and (5.10) to find that,

[TABLE]

Thus, as $t\searrow 0$, we have:

[TABLE]

that is,

[TABLE]

On the other hand, from (5.17) we easily check that,

[TABLE]

and therefore, as $t\searrow 0$:

[TABLE]

As a consequence, from (5.22) and the Moser-Trudinger inequality, as $t\to 0^{+}$, we find:

[TABLE]

with suitable positive constants $C_{1}$ and $C_{2}$.

Thus, we obtain:

[TABLE]

with a suitable constant $C>0$, independent of $r>0$. So by letting $r\searrow 0$, we get that $K(p_{0})>0$ and (5.21) is proved.

At this point, for any sequence $t_{n}\searrow 0$, we can apply Theorem D for the sequence $v_{2,t_{n}}$. In view of (5.18) and (5.19), we know that $v_{2,t_{n}}$ can admit exactly one blow-up point at $p_{0}$ with $K(p_{0})\not=0$. Therefore (3.6) must hold with $m=1<g-1$, and consequently properties (i)-(iii) must hold for $v_{2,t_{n}}$.

Since this holds along any sequence $t_{n}\searrow 0$, we obtain the desired conclusion. ∎

5.2. Blow-up analysis when $g=2$

When the surface is of genus $g=2$, the asymptotic behavior of $v_{2,t}$ is governed by the extremal properties of the Moser-Trudinger functional $\mathcal{J}$ in (1.5) (see for instance [Tru67, Mos71, Aub82]). Indeed, the goal of this subsection is to prove the following:

Theorem 5.3**.**

Let $S$ be of genus $g=2$. Then, as $t\searrow 0$, we have:

[TABLE]

[TABLE]

[TABLE]

where $\mathcal{J}$ is the Moser-Trudinger functional defined in (1.5), and $E=\{w\in H^{1}(S):\int_{S}w(z)\ dA=0\}$.

Furthermore, by setting

[TABLE]

then the following alternative holds:

1. (i)

either*, $\mathcal{J}$ attains its infimum on $E$, and along a subsequence $t=t_{n}\to 0$, as $n\to\infty$, we have:*

[TABLE]

and

[TABLE]

with $w_{0}$ satisfying the following equation on $S$:

[TABLE] 2. (ii)

or*, the functional $\mathcal{J}$ does not attain its infimum on $E$, and along a subsequence $t=t_{n}\to 0$, as $n\to\infty$, for*

[TABLE]

we have:

[TABLE]

and

[TABLE]

weakly in the sense of measure, and

[TABLE]

uniformly in $C_{loc}^{2,\beta}(S\backslash\{p_{0}\})$, where $0<\beta<1$, with the blow-up point $p_{0}\in S$ satisfying:

[TABLE]

and in particular, $\alpha(p_{0})\not=0$.

Remark 5.4**.**

Clearly, if we knew the uniqueness of the minimum of the Moser-Trudinger functional $\mathcal{J}$ on $E$ (when attained), or of the maximum point of the function $4\pi\gamma(p,p)+\log K(p)$, we could claim the convergence above as $t\to 0$, not only along a subsequence $t=t_{n}\to 0$ as $n\to\infty$.

Concerning the existence of a global minimum of $\mathcal{J}$ in $E$, we briefly recall the work of Ding-Jost-Li-Wang ([DJLW97]) and Nolasco-Tarantello ([NT98]):

Lemma 5.5**.**

We have:

[TABLE]

and the infimum is attained if (5.34) holds with a strict inequality.

Proof.

See [DJLW97, NT98]. ∎

On the basis of Lemma  5.5, the existence of a global minimum for the extremal problem:

[TABLE]

was ensured by the authors in [DJLW97, NT98] under the following sufficient condition:

[TABLE]

with $p_{0}$ satisfying (5.33), and $\kappa$ the Gauss curvature of $(S,\sigma)$. See also [Tar08].

For our geometrical problem, we have $K=\frac{|\alpha|^{2}}{det(g_{\sigma})}$, with $g_{\sigma}$ the hyperbolic metric, and $\alpha\in Q(\sigma)$ a holomorphic quadratic differential on the Riemann surface $(S,\sigma)$. So for any $p_{0}\in S$ with $\alpha(p_{0})\not=0$, we have $\kappa(p_{0})=-1$, $|S|_{\sigma}=4\pi$ (note that $g=2$), and

[TABLE]

Therefore we see that both sides of (5.35) are equal to $-4$, and in this sense we just “missed” to satisfy this sufficient condition (5.36).

Proof.

We first apply (2.2) to the solution $v_{2,t}$ which (for $g=2$) implies that (3.6) must hold with $m=g-1=1$. Therefore we have, as $t\searrow 0$,

[TABLE]

As before, we write $v_{2,t}=w_{t}+\mathtt{c}_{t}$, and (by Jensen’s inequality) we find:

[TABLE]

This establishes (5.23) and (5.24). Notice that we are now in the situation described by part (ii) of Theorem D.

In order to establish (5.25), we use (5.23) and $g=2$, to conclude that the mean value $\mathtt{c}_{t}$ of $v_{2,t}$ must satisfy (2.4) with the “plus” sign. In other words,

[TABLE]

So, we can use (5.37) to write

[TABLE]

Consequently,

[TABLE]

and we derive the lower bound:

[TABLE]

To obtain the reversed inequality, we will construct some “optimal” path. To this purpose, for any fixed $w\in E$, we find $t_{w}>0$ sufficiently small, such that:

[TABLE]

So for every $t\in(0,t_{w})$, we can define

[TABLE]

Also set, corresponding to the stable solution $v_{1,t}$:

[TABLE]

and

[TABLE]

We define the following path:

[TABLE]

with $\tilde{C}_{t}>0$ fixed sufficiently large (depending on $w$), to ensure that,

[TABLE]

Therefore $\Gamma_{t,w}\in\mathcal{P}_{t}$, the path space defined in (4.27). Since

[TABLE]

we readily check that,

[TABLE]

with a suitable constant $C(w)>0$ depending on $w$ only.

On the other hand, for $s\in[\frac{1}{2},1]$ and $t\in(0,t_{w})$, we have:

[TABLE]

So, by observing that $v=w+\mathtt{c}_{t}^{+}(w)$ satisfies the integral identity (2.2), we can use (5.41) to show that

[TABLE]

So from (5.45) and (5.46), for $t>0$ sufficiently small, we find:

[TABLE]

As a consequence, we get:

[TABLE]

Since (5.48) holds for every $w\in E$, and using (5.40), we establish (5.25). Actually, from (5.39) and (5.48), we see that,

[TABLE]

Next we wish to show that $w_{t}$ satisfies the “compactness” alternative in part (ii) of Theorem D (along a subsequence $t=t_{n}\searrow 0$) if and only if $\mathcal{J}$ attains its infimum in $E$.

To this purpose, we fix $w\in E$, and as before set $t_{w}>0$ sufficiently small to ensure that,

[TABLE]

We set a function

[TABLE]

for $A\in[0,4\pi^{2}]$. Clearly this is a monotone increasing function of $A$ in $[0,4\pi^{2}]$.

From (5.47) and (5.38), it follows that, for $\forall w\in E$, and $\forall t\in(0,t_{w})$, there holds:

[TABLE]

Therefore if we assume the functional $\mathcal{J}$ attains its infimum at $w_{0}$, that is,

[TABLE]

then we can use $w=w_{0}$ in (5.51) to conclude that,

[TABLE]

Now we use the fact that $f$ is increasing in $A$ and from (5.53) to derive that,

[TABLE]

with $\fint_{S}e^{-w_{t}}\geq 1$, by Jensen’s inequality.

Therefore, for suitable $C_{1}>0$, we have,

[TABLE]

Using $v_{2,t}=w_{t}+\mathtt{c}_{t}$ in Lemma  2.4, we find that,

[TABLE]

for suitable $C_{2}>0$, and moreover $w_{t}$ is uniformly bounded in $W^{1,q}(S)$, $1<q<2$. Now from (5.37), (5.54) and (5.55), we can deduce that:

[TABLE]

with a suitable constant $C_{3}>1$.

In addition, from (5.49), with $w=w_{0}$, we get

[TABLE]

with a suitable constant $C_{4}>0$.

So in case the functional $\mathcal{J}$ attains its infimum in $E$, then we can use the estimates in (5.56) and (5.57) together with elliptic estimates and well known regularity theory, to conclude that, $w_{t}$ is uniformly bounded in $C^{2,\beta}(S)$-norm, with $0<\beta<1$, for any $t\in(0,t_{w_{0}})$. Consequently, along a subsequence $t_{n}\searrow 0$, $w_{n}:=w_{t_{n}}$ satisfies the “compactness” property of part (ii) in Theorem D. In other words, (5.27) holds with $w_{0}$ satisfying (5.28) and (5.29).

Next suppose the functional $\mathcal{J}$ does not attain its infimum in $E$. Therefore, $w_{t}$ can not satisfy the “compactness” property in part (ii) in Theorem D. Consequently, by (5.23) we know that, along a sequence $t_{n}\searrow 0$, the sequence $w_{n}:=w_{t_{n}}$ must admit one $(m=1)$ blow-up point $p_{0}\in S$, satisfying (5.30), (5.31), and (5.32). So we are left to show that (5.33) holds. To this purpose, from (5.49), we know that $w_{n}$ defines a minimizing sequence for $\mathcal{J}$ in $E$ and $\max\limits_{S}w_{n}\to+\infty$, i.e., blow-up occurs. Therefore, we can use for $w_{n}$, the estimates detailed in [DJLW97] and [NT98] for any blow-up minimizing sequences of $\mathcal{J}$, to show that,

[TABLE]

On the other hand, when $\mathcal{J}$ does not attain its infimum, we also know that,

[TABLE]

See Lemma  5.5 and [DJLW97]. Now (5.33) follows immediately from (5.58) and (5.59). ∎

6. Prescribing extrinsic curvature

In this section, we wish to investigate the possibility to obtain a minimal immersion of $S$ into a hyperbolic three-manifold with prescribed total extrinsic curvature. Namely, for given $\rho\in(0,4\pi(g-1))$, we require that for the induced metric $g_{0}$ we have:

[TABLE]

6.1. Main result and three lemmata

Our main result is the following:

Theorem E**.**

Fixing a conformal structure $\sigma\in T_{g}(S)$, and a holomorphic quadratic differential $\alpha\in Q(\sigma)$, and $\rho\in(0,4\pi(g-1))\backslash\{4\pi m,m=2,\cdots,g-2\}$, there exists a constant $t_{\rho}\in(0,\tau_{0}]$ ($\tau_{0}=\tau_{0}(\sigma,\alpha)>0$ given in Theorem  0.1), such that $S$ admits a minimal immersion of data $(\sigma,t_{\rho}\alpha)$ into some hyperbolic three-manifold, with corresponding total extrinsic curvature satisfying (6.1).

In order to establish this result, we need to provide a solution $v_{\rho}$ for the problem $(1)_{t_{\rho}}$, for some $t_{\rho}\in(0,\tau_{0}]$ satisfying:

[TABLE]

To this purpose, we recall from §3.1 the Mean Field formulation of the problem $(1)_{t}$, as described in Lemma  3.1. Then for given $\rho\in(0,4\pi(g-1))$ we need to find a solution $w$ of the equation (3.4), that is:

[TABLE]

We call this equation, (namely (3.4)), the problem $(3)_{\rho}$.

To this end, we start with the following observation:

Lemma 6.1**.**

If $w$ solves the problem $(3)_{\rho}$ with $\rho\in(0,4\pi(g-1))$, then:

[TABLE]

Proof.

We recall from Lemma  3.1 that when $w$ solves problem $(3)_{\rho}$, then we define:

[TABLE]

and

[TABLE]

and we see that $v_{\rho}=w+\mathtt{c}_{\rho}$ is a solution to problem $(1)_{t_{\rho}}$. Since we have:

[TABLE]

and $v_{\rho}>0$, then, $\forall\ z\in S$:

[TABLE]

and this establishes (6.3). ∎

As a consequence of Corollary  3.5, we know that, for any $\rho\in(4\pi(m-1),4\pi m)$, $m=\{1,\cdots,g-1\}$, the Leray-Schauder degree of the Fredholm operator associated to the problem $(3)_{\rho}$ is well-defined, and its value only depends on $m$. To be more precise, we recall that the Laplace-Beltrami operator $\Delta=\Delta_{g_{\sigma}}$ is invertible on the space $E$. We denote by:

[TABLE]

its (smooth) inverse. Thus each solution to the problem $(3)_{\rho}$ corresponds to a zero of the following operator:

[TABLE]

with

[TABLE]

and

[TABLE]

Therefore, in view of Lemma  6.1, there exists a suitable constant $C>0$ (independent of $\rho$), such that if $w\in E$ is a solution of problem $(3)_{\rho}$, that is $F_{\rho}(w)=0$, then $\|B_{\rho}(w\|\leq C$.

As a consequence, for any $\rho\in(4\pi(m-1),4\pi m)$, with $m=\{1,\cdots,g-1\}$, we find a radius $R_{\rho}$ sufficiently large, such that, for each $t\in[0,1]$ it is well-defined at zero the Leray-Schauder degree $d_{\rho,t}$ of the operator

[TABLE]

in the ball $B_{R_{\rho}}=\{w\in E:\|w\|\leq R_{\rho}\}$. Moreover, by the homotopy invariance of the Leray-Schauder degree, we have

[TABLE]

In particular,

[TABLE]

where $d_{\rho,0}$ is the Leray-Schauder degree of the operator

[TABLE]

whose zeroes correspond to solutions of the following problem:

[TABLE]

Actually for every $\rho\not\in 4\pi\mathbb{N}$, the Leray-Schauder degree of the operator $F_{\rho}^{0}$ has been computed by Chen-Lin ([CL03, CL15]), exactly when the weight function $K$ admits isolated zeros (which is our case) $\{q_{1},\cdots,q_{N}\}$, each with integer multiplicity $\nu(q_{j}),\ 1\leq j\leq N$. More precisely we have the following:

Lemma 6.2**.**

([CL15]) If $\nu(q_{j})\in\mathbb{N}\ \forall 1\leq j\leq N$, and the genus of the surface is greater than zero, then $d_{\rho,0}>0$.

Proof.

See Corollary 1.2 in [CL15]. ∎

6.2. Proof of Theorem E

We now complete the proof of Theorem E:

Proof.

Since in our case, the weight function $K(z)=\|\alpha\|_{\sigma}^{2}$ with $\alpha\in Q(\sigma)$ a holomorphic quadratic differential on $S$, we know that $\alpha$ admits isolated zeroes of integer multiplicity and total number (counting multiplicity) equal to $4(g-1)$. Thus, we can apply Lemma  6.2 together with (6.10) to conclude that, for every $\rho\in(0,4\pi(g-1))\backslash\{4\pi m,m=1,\cdots,g-2\}$, the Leray-Schauder degree $d_{\rho}>0$. In other words, for such range of $\rho$’s, we know that the problem $(3)_{\rho}$ admits at least one solution. To complete the proof, we need to show that, when $g\geq 3$, then we have the existence of a solution for problem $(3)_{\rho}$ also when $\rho=4\pi$.

To this purpose, we once again exploit the work of Chen-Lin in [CL10, CL02]. We take a sequence $w_{n}$ of the solutions to problem $(3)_{\rho_{n}}$, with $4\pi m\not=\rho_{n}\to 4\pi m$, for some $m\in\{1,\cdots,g-2\}$. We assume that, as $n\to+\infty$, the following holds:

[TABLE]

and

[TABLE]

Indeed, in case $w_{n}$ was uniformly bounded in $S$, then by elliptic estimates (along a subsequence) it would converge to a solution to problem $(3)_{\rho=4\pi m}$, and for $m=1$ we would obtain our solution in this way. Thus, we assume (6.12) and we want to establish a sign for the quantity $\rho_{n}-4\pi m$. This delicate task has been carried out by Chen-Lin in [CL10, CL02] for the sequence $z_{n}=w_{n}-\zeta_{n}$, satisfying (3.17) and (3.18), with $\zeta_{n}$ defined in (3.14) and satisfying (along a subsequence):

[TABLE]

with $\zeta_{0}$ the unique solution for:

[TABLE]

From (6.12) we know that, $\max\limits_{S}z_{n}\to+\infty$. Therefore, by using Theorem  3.2, $z_{n}$ must admit a finite number of blow-up points, say $\{p_{1},\cdots,p_{s}\}\subset S$, for which (3.20) holds with $m=\sum\limits_{j=1}^{s}(1+n(p_{j}))$ and

[TABLE]

If we further assume that these blow-up points are not zeroes of the weight function $K=\|\alpha\|_{\sigma}^{2}$, that is $\alpha(p_{j})\not=0,\ \forall j=1,\cdots,s$; then $n(p_{j})=0$ and $m=s$. In this situation, Chen-Lin in [CL15] were able to control the exact decay to zero of the quantity: $\rho_{n}-4\pi m$. In particular they showed that, the sign of $\rho_{n}-4\pi m$ is the same of the following quantity:

[TABLE]

where $d_{j}$’s are suitable (positive) constants, $h^{*}=Ke^{\zeta_{0}}$, and $\kappa$ is the Gauss curvature of $S$. Take into account also that the expression (6.15) was given in [CL15] by formulae $(2.3)$ and $(2.10)$, written under the normalization $|S|=1$. Now, for $p\in S:\alpha(p)\not=0$, by means of (6.14), we compute:

[TABLE]

Therefore, we may conclude that, if $K$ (and hence $\alpha$) does not vanish at the blow-up points of $w_{n}$, then for $n$ sufficiently large, we have: $\rho_{n}-4\pi m<0$. That is, blow-up can only occur from the “right”.

Since for $m=1$, the solutions to problem $(3)_{\rho_{n}}$ with $\rho_{n}\to 4\pi$ can admit only one blow-up point $p_{0}\in S$ which must satisfy $K(p_{0})\not=0$. Therefore, we can use the information above, to see that for $\rho_{n}>4\pi$ and $\rho_{n}\searrow 4\pi$, the corresponding solution $w_{n}$ cannot blow-up, and so (along a subsequence) it converges to the desired solution of $(3)_{\rho=4\pi}$. ∎

We conclude the section with two remarks.

Remark 6.3**.**

When the sequence of solutions $w_{n}$ to problem $(3)_{\rho_{n}}$ with $\rho_{n}\to 4\pi m$, blows up at a zero of $K=\|\alpha\|_{\sigma}^{2}$, we suspect that similar information about the sign of the quantity $\rho_{n}-4\pi m$ should hold. This is confirmed by the more involved analysis developed in [CL10], where the authors provide sharp estimates about the behavior of the sequence $z_{n}$ of (3.18), (3.17), which blows up at a zero of the weight function $K$, but only when such a zero is of non-integer multiplicity.

Remark 6.4**.**

Finally we note that, in view of (3.7), (3.8), by choosing $\alpha\in Q(\sigma)$ with zeroes of multiplicity greater than $g-2$, we can always guarantee that blow-up never occurs at its zeroes.