Upper Measure Bounds of Nodal Sets of Solutions to the Bi-Harmonic Equations on $C^{\infty}$ Riemannian Manifolds

TL;DR

This paper establishes upper bounds for the measure of nodal sets of bi-harmonic functions and eigenfunctions on smooth Riemannian manifolds, using frequency functions and doubling conditions to quantify their size.

Contribution

It introduces monotonicity formulas for frequency functions of bi-harmonic functions and derives new bounds on nodal set measures based on these functions.

Findings

Upper bounds for nodal set measures depend on the frequency function N.

Nodal set measure is controlled by N^α for small balls.

Eigenfunction nodal sets are bounded by a power of the eigenvalue λ.

Abstract

In this paper, we consider the nodal set of a bi-harmonic function $u$ on an $n$ dimensional $C^{\infty}$ Riemannian manifold $M$, that is, $u$ satisfies the equation $\triangle_M^2u=0$ on $M$, where $\triangle_M$ is the Laplacian operator on $M$. We first define the frequency function and the doubling index for the bi-harmonic function $u$, and then establish their monotonicity formulae and doubling conditions. With the help of the smallness propagation and partitions, we show that, for some ball $B_r(x_0)\subseteq M$ with $r$ small enough, an upper bound for the measure of nodal set of the bi-harmonic function $u$ can be controlled by $N^\alpha$, that is, \mathcal{H}^{n-1}\left(\left\{x\in B_{r/2}(x_0)|u(x)=0\right\}\right)\leq CN^{\alpha}r^{n-1}, where $N=\max\left\{C_0,N(x_0,r)\right\}$, $\alpha$, $C$ and $C_0$ both are positive constants depending only on $n$ and $M$. Here $N(x_0,r)$ is the frequency function of $u$ centered at $x_0$ with radius $r$. Furthermore, we derive that an upper measure for nodal sets of eigenfunctions of the bi-harmonic operator on a $C^{\infty}$ compact Riemannian manifold without boundary can be controlled by $\lambda^\beta$ for the corresponding eigenvalue $\lambda^2$ and some positive constant $\beta$.