On constant mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds

TL;DR

This paper investigates the moduli space of constant mean curvature 1-immersions of surfaces into hyperbolic 3-manifolds, analyzing blow-up phenomena and limits, with new conditions involving the Kodaira map for genus 2 and higher.

Contribution

It refines the understanding of blow-up limits for CMC 1-immersions, especially for genus 2 and 3 surfaces, introducing new conditions involving the Kodaira map and improving asymptotic analysis.

Findings

For genus 2, blow-up occurs at a single point, with conditions involving the Kodaira map on Weierstrass points.

For higher genus, a new replacement for the Kodaira map is identified on the space of divisors.

Enhanced asymptotic analysis applicable to genus 3, indicating behavior in the general case.

Abstract

Motivated by the work of Bryant on constant mean curvature (CMC) $1$-immersions of surfaces into the hyperbolic space H^3 and after the results of Tarantello (2023), we pursue a possible parametrization for the moduli space of (CMC) 1-immersions of a surface S (closed, orientable and of genus >1) into hyperbolic 3-manifolds. Those immersions enter as "critical" object in our analysis. In fact, they can be attained only as limits of the (CMC) c-immersions (as c tends to 1), obtained in Huang-Lucia-Tarantello (2022), for |c|<1. However, such passage to the limit can be prevented by possible blow-up phenomena, so that the pullback metrics of the (CMC) c-immersions may yield (at the limit) to a singular metric with conical singularities at finitely many points (the blow-up points). In case of genus g=2, blow up can occur only at a single point, and in Tarantello (2023) it was shown how it could be prevented and the passage to the limit ensured in terms of the Kodaira map. In this note we sharpen this result and for genus g=2, we obtaina condition (we believe sharp) which involves only the Kodaira map on the six Weierstrass points. In addition we tackle the case of higher genus, where multiple blow-up points occur. In this case, we need to identify a suitable replacement of the Kodaira map, now defined on the space of non-zero effective divisors. More importantly, we need to improve in a substantial way the asymptotic analysis of Tarantello (2023) limited to the case of "blow-up" with minimal mass. In this direction we give a contribution which best applies to the case of genus g=3, but also provides a relevant step and a convincing indication on what should happen in the general case.