[Paper Review] Compactness and Rigidity of $\lambda$-Surfaces

Ao Sun|arXiv (Cornell University)|Apr 25, 2018

Geometric Analysis and Curvature Flows14 references3 citations

TL;DR

This paper establishes a compactness theorem for $λ$-surfaces in $×^3$ with uniform genus, area growth, and $\lambda$-bound, generalizing Colding-Minicozzi's result for self-shrinkers. It proves a rigidity theorem showing that any convex $λ$-surface with diameter less than $D$ must be a sphere when $\u03bb > \delta_D$, with $\delta_D < 0$, extending known results to the negative $\u03bb$ regime using new analytic techniques.

ABSTRACT

In this paper we develop the compactness theorem for $\\lambda$-surface in $\\mathbb R^3$ with uniform $\\lambda$, genus, and area growth. This theorem can be viewed as a generalization of Colding-Minicozzi's compactness theorem for self-shrinkers in $\\mathbb R^3$. As an application of this compactness theorem, we prove a rigidity theorem for convex $\\lambda$-surfaces.

Motivation & Objective

• To establish a compactness theorem for $λ$-surfaces in $×^3$ under uniform bounds on genus, area growth, and $\lambda$.
• To generalize Colding-Minicozzi's compactness result for self-shrinkers ($\lambda = 0$) to general $λ$-surfaces.
• To prove a rigidity theorem for convex $λ$-surfaces, showing they must be spheres when $λ > \delta_D$ for some $\delta_D < 0$, extending prior results to negative $λ$.
• To address the challenge of $λ < 0$ case, where $λ$-surfaces behave differently than for $λ \geq 0$, requiring new techniques.

Proposed method

• Develop a compactness theorem for $λ$-surfaces using uniform bounds on genus, area growth, and $|\lambda| \leq \Lambda$, generalizing Colding-Minicozzi's approach.
• Apply the compactness theorem to sequences of convex $λ$-surfaces with bounded diameter to extract subsequential limits.
• Use the implicit function theorem on the linearized $λ$-surface operator $L = \Delta_{S^2_r} + \frac{1}{2} + \frac{2}{r^2}$ at $r=2$, where $L = \Delta_{S^2_2} + 1$, to show uniqueness of solutions near $\lambda = 0$.
• Employ the Fredholm alternative to prove that the linearized operator is a local isomorphism, enabling local uniqueness of solutions via the implicit function theorem.
• Use the area growth bound $\text{Area}(B_R(x_0) \cap \Sigma) \leq 4\pi R^2$ for convex surfaces to satisfy the area condition in the compactness theorem.
• Apply maximum principle and convexity to rule out self-touching or multiple-sheeted limits, ensuring smooth, embedded convergence to a sphere when $\lambda \to 0^+$.

Experimental results

Research questions

• RQ1 Can a compactness theorem for $λ$-surfaces be established under uniform bounds on genus, area growth, and $|\lambda|$?
• RQ2 Does the rigidity of convex $λ$-surfaces extend to negative $λ$ values, as it does for $\u03bb \geq 0$?
• RQ3 What is the sharp threshold $\delta_D < 0$ such that all convex $λ$-surfaces with diameter $< D$ are spheres when $\u03bb > \delta_D$?
• RQ4 How do $λ$-surfaces behave differently for $\u03bb < 0$ compared to $\u03bb \geq 0$, and what new techniques are needed to analyze them?
• RQ5 Can the compactness theorem be used to classify convex $λ$-surfaces via convergence to self-shrinkers or embedded spheres?

Key findings

• A compactness theorem is established for $λ$-surfaces in $×^3$ with uniform genus, area growth, and $|\lambda| \leq \Lambda$, generalizing Colding-Minicozzi’s result for self-shrinkers.
• The limit of a sequence of $λ$-surfaces may be either a smooth self-touching $λ$-surface or a self-shrinker with multiplicity 2, depending on the presence of neck pinching points.
• For convex $λ$-surfaces with diameter less than $D$, there exists $\delta_D < 0$ such that any such surface with $\lambda > \delta_D$ must be a sphere of radius $r = \sqrt{\lambda^2 + 4} - \lambda$, extending rigidity to negative $λ$.
• The proof relies on the implicit function theorem applied to the linearized $λ$-surface operator at $\lambda = 0$, where the operator $L = \Delta_{S^2_2} + 1$ is invertible, ensuring local uniqueness of solutions.
• The area growth bound $\text{Area}(B_R(x_0) \cap \Sigma) \leq 4\pi R^2$ holds for all convex surfaces in $×^3$, even non-compact ones, enabling application of the compactness theorem.
• The existence of $\delta_D < 0$ is proven by contradiction: assuming no such $\delta_D$ exists leads to a sequence of non-spherical convex $λ_i$-surfaces converging to a sphere, contradicting the uniqueness from the implicit function theorem.