Volume growth and puncture repair in conformal geometry

A complete conformal metric of preassigned negative Gaussian curvature for a punctured hyperbolic Riemann surface

Let $h$ be a complete metric of Gaussian curvature $K_0$ on a punctured Riemann surface of genus $g \geq 1$ (or the sphere with at least three punctures). Given a smooth negative function $K$ with $K=K_0$ in neighbourhoods of the punctures we prove that there exists a metric conformal to $h$ which attains this function as its Gaussian curvature for the punctured Riemann surface. We do so by minimizing an appropriate functional using elementary analysis.

Conformal Structure of Minimal Surfaces with Finite Topology

In this paper, we show that a complete embedded minimal surface in $\Real^3$ with finite topology and one end is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic to that of the helicoid. More precisely, if $g$ is the stereographic projection of the Gauss map, then in a neighborhood of the puncture, $g(p) = \exp(iαz(p) + F(p))$, where $α\in \Real$, $z=x_3+ix_3^*$ is a holomorphic coordinate defined in this neighborhood and $F(p)$ is holomo…

The geometry of closed conformal vector fields on Riemannian spaces

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T. K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to …

Morse 指标、拓扑及具有非紧自由边界的极小曲面的端部

Morse index, topology, and ends of minimal surfaces with noncompact free boundary

本文为具有非紧自由边界的完备双侧极小曲面建立 Morse 指标下界，结合局部能量恒等式与 Riemann--Roch 计数，并处理内部端与边界端的截断问题，推广了紧边界情形的结果。

第六次GJMS方程奇异解的紧致性

Compactness of singular solutions to the sixth order GJMS equation

研究第六次GJMS方程奇异解的紧致性，通过necksize概念和Gromov-Hausdorff度量分析，结合上界和下界估计，证明子序列收敛性。

Morse Index Stability for Critical Points to Conformally invariant Lagrangians

We prove the upper-semi-continuity of the Morse index plus nullity of critical points to general conformally invariant Lagrangians in dimension 2 under weak convergence. Precisely we establish that the sum of the Morse indices and the nullity of an arbitrary sequence of weakly converging critical points to a general conformally invariant Lagrangians of maps from an arbitrary closed surface into an arbitrary closed smooth manifold passes to the limit in the following sense : it is asymptotically bounded from above by the sum of the Morse indices plus the nullity of the weak limit and the bubble…