[Paper Review] The rigidity theorems of self shrinkers

Qi Ding, Y. L. Xin|arXiv (Cornell University)|May 25, 2011

Geometric Analysis and Curvature Flows12 references4 citations

TL;DR

This paper establishes rigidity theorems for self-shrinkers in Euclidean space by analyzing the squared norm of the second fundamental form using weighted $L^2$-estimates and Sobolev-type inequalities under the $\mathcal{L}$-operator. It proves that if the squared norm $S$ of the second fundamental form satisfies certain pointwise or integral conditions, then $|\nabla B| \equiv 0$, implying the self-shrinker is totally geodesic or a generalized sphere.

ABSTRACT

By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We prove rigidity results for squared norm of the second fundamental form of self-shrinkers, either under point-wise conditions or under integral conditions.

Motivation & Objective

• To establish pointwise and integral rigidity conditions for self-shrinkers based on the squared norm $S$ of the second fundamental form.
• To extend gap phenomena results from minimal submanifolds to self-shrinkers, particularly the second gap for $S$.
• To classify self-shrinker surfaces in $\mathbb{R}^3$ with constant $S$.
• To prove that under specific integral or pointwise bounds on $S$, the second fundamental form must be parallel ($|\nabla B| \equiv 0$).

Proposed method

• Utilizes the $\mathcal{L}$-operator $\mathcal{L} = \Delta - \frac{1}{2}\langle X, \nabla(\cdot) \rangle$, which is self-adjoint with respect to the measure $\rho d\mu = e^{-|X|^2/4} d\mu$.
• Applies weighted $L^2$-estimates and integration by parts on the $\mathcal{L}$-Laplacian of $S = |B|^2$ to derive inequalities involving $|\nabla B|^2$, $|\nabla S|^2$, and $|\nabla^2 B|^2$.
• Employs the Simons-type formula for $\mathcal{L}S$ and estimates involving $S(S - 1/2)$ and $|\nabla B|$, leveraging Cauchy-Schwarz and Young’s inequality.
• Uses Sobolev-type inequalities and direct integral estimates to control $|\nabla B|^3$ and $|\nabla S|^2$ terms via $S$ and $|\nabla B|^2$.
• Applies a critical test with $\delta = 0.011$ and $\theta = 1/2$ to show that the coefficient of $\int |\nabla B|^2 \rho$ becomes positive definite, forcing $|\nabla B| \equiv 0$.
• Derives a classification of $2$-dimensional self-shrinkers in $\mathbb{R}^3$ with constant $S$, showing they are either planes or round spheres.

Experimental results

Research questions

• RQ1 Under what pointwise or integral conditions on $S = |B|^2$ does a self-shrinker become totally geodesic?
• RQ2 Can the second gap phenomenon for $S$ in minimal submanifolds be extended to self-shrinkers in Euclidean space?
• RQ3 What is the classification of self-shrinker surfaces in $\mathbb{R}^3$ with constant squared norm of the second fundamental form?
• RQ4 How do weighted $L^2$-estimates under the $\mathcal{L}$-operator control the growth of $|\nabla B|$?
• RQ5 Is the gap in $S$ sharp for self-shrinkers, and can it be proven via direct integral estimates without compactness?

Key findings

• For self-shrinkers in $\mathbb{R}^{m+n}$, if $S = |B|^2$ satisfies $S \leq 1/2$ pointwise, then $|\nabla B| \equiv 0$, implying the submanifold is totally geodesic.
• Under the integral condition $\int_M S(S - 1/2) \rho \, d\mu < \infty$ and specific parameter choices ($\delta = 0.011$, $\theta = 1/2$), the inequality forces $|\nabla B| \equiv 0$.
• The second gap for $S$ in self-shrinkers is confirmed to be sharp: if $S \in (1/2, 1)$, then $|\nabla B| \equiv 0$ under the derived integral estimates.
• For $2$-dimensional self-shrinkers in $\mathbb{R}^3$ with constant $S$, the only possibilities are the plane ($S = 0$) or the round sphere ($S = 1$), as shown in Theorem 4.2.
• The proof establishes that $|\nabla B| \equiv 0$ via a contradiction argument using a positive definite coefficient in the $L^2$-estimate of $|\nabla B|^2$, achieved by optimizing $\epsilon$ and $\delta$.
• The method successfully extends techniques from minimal submanifold theory to self-shrinkers, proving rigidity under both pointwise and integral conditions on $S$.