[Paper Review] Two-point functions and their applications in geometry

Simon Brendle|arXiv (Cornell University)|Feb 7, 2014

Geometric Analysis and Curvature Flows33 references3 citations

TL;DR

This paper presents a refined application of the maximum principle using two-point functions to analyze geometric flows and minimal surfaces, particularly proving Lawson’s 1970 conjecture that the only embedded minimal torus in $S^3$ is the Clifford torus. By constructing a carefully chosen function $Z_\kappa$ depending on two points on a surface and analyzing its Laplacian and mixed derivatives, the authors derive a differential inequality that leads to a contradiction unless the surface is the Clifford torus, leveraging Bony’s strict maximum principle for degenerate elliptic equations.

ABSTRACT

The maximum principle is one of the most important tools in the analysis of geometric partial differential equations. Traditionally, the maximum principle is applied to a scalar function defined on a manifold, but in recent years more sophisticated versions have emerged. One particularly interesting direction involves applying the maximum principle to functions that depend on a pair of points. This technique is particularly effective in the study of problems involving embedded surfaces. In this survey, we first describe some foundational results on curve shortening flow and mean curvature flow. We then describe Huisken's work on the curve shortening flow where the method of two-point functions was introduced. Finally, we discuss several recent applications of that technique. These include sharp estimates for mean curvature flow, as well as the proof of Lawson's 1970 conjecture concerning minimal tori in S^3.

Motivation & Objective

• To extend the classical maximum principle to functions of two points on a manifold, enabling stronger rigidity results in geometric analysis.
• To resolve Lawson’s 1970 conjecture on the uniqueness of embedded minimal tori in $S^3$.
• To develop a framework using two-point functions for analyzing mean curvature flow and minimal surfaces in curved ambient spaces.
• To establish sharp estimates for mean curvature flow and classify constant mean curvature surfaces in $S^3$.

Proposed method

• Introduce a two-point function $Z_\kappa(x,y)$ involving the second fundamental form and inner products of position vectors on a surface in $S^3$.
• Compute the Laplacian of $Z_\kappa$ with respect to both $x$ and $y$ variables, using the Codazzi equations and geometric identities.
• Derive a differential inequality involving second derivatives and mixed derivatives, showing non-positivity under certain curvature conditions.
• Apply Bony’s strict maximum principle to a variant of the inequality valid away from the diagonal, proving the set where $Z_\kappa = 0$ is open and non-empty.
• Use Taylor expansion of the two-point function near the diagonal to deduce that $\nabla|A| = 0$, implying constant mean curvature and symmetry.
• Leverage Lawson’s theorem on surfaces with constant $|A|$ to conclude that the surface must be congruent to the Clifford torus.

Experimental results

Research questions

• RQ1 Can the maximum principle be generalized to functions depending on two points to prove rigidity results in geometric analysis?
• RQ2 Is the Clifford torus the only embedded minimal torus in $S^3$?
• RQ3 What conditions on the second fundamental form and curvature lead to rigidity in minimal surfaces via two-point function techniques?
• RQ4 Can two-point function methods be extended to constant mean curvature or Weingarten surfaces in $S^3$?

Key findings

• Lawson’s 1970 conjecture is confirmed: the only embedded minimal torus in $S^3$ is the Clifford torus.
• The two-point function method yields a contradiction unless $|A|$ is constant, implying the surface must be the Clifford torus.
• The proof establishes that $\nabla|A| = 0$ at every point via Taylor expansion of the two-point function near the diagonal.
• The method extends to constant mean curvature tori in $S^3$, showing they must be rotationally symmetric if immersed in the sense of Alexandrov.
• The differential inequality derived from the Laplacian of $Z_\kappa$ is non-positive, leading to a contradiction unless the surface is the Clifford torus.
• The technique provides sharp estimates for mean curvature flow and applies to a class of Weingarten tori in $S^3$.