[Paper Review] Volume estimate about shrinkers

Xu Cheng, Detang Zhou|arXiv (Cornell University)|Jun 24, 2011

Geometric Analysis and Curvature Flows6 references4 citations

TL;DR

This paper establishes a precise volume growth estimate for complete noncompact Riemannian manifolds equipped with a potential function satisfying shrinking soliton-type conditions. By analyzing the weighted volume and level set growth of such functions, the authors derive sharp polynomial volume bounds for self-shrinkers in Euclidean space and gradient shrinking Ricci solitons, proving the equivalence of properness, polynomial volume growth, and finite weighted volume for self-shrinkers.

ABSTRACT

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space.

Motivation & Objective

• To derive a general volume growth estimate for complete noncompact Riemannian manifolds with a potential function satisfying specific differential inequalities.
• To apply the general estimate to self-shrinkers in Euclidean space and gradient shrinking Ricci solitons, yielding explicit volume growth bounds.
• To establish the equivalence of properness, polynomial volume growth, and finite weighted volume for immersed self-shrinkers in Euclidean space.
• To re-interpret and rephrase the Colding-Minicozzi compactness theorem using the finite weighted volume condition.
• To provide a new, independent proof of the Euclidean volume growth estimate for gradient shrinking solitons using potential function techniques.

Proposed method

• Define the weighted Laplacian $\Delta_f u = \Delta u - \langle \nabla f, \nabla u \rangle$ and the weighted measure $e^{-f} dv$.
• Introduce the level sets $D_r = \{x \in M : 2\sqrt{f} < r\}$ and analyze their volume $V(r)$ and weighted volume $V_f(r)$.
• Establish the key estimate $V(r) \leq C r^{2k}$ for $r \geq 1$ under the assumptions $|\nabla f|^2 \leq f$ and $\Delta_f f + f \leq k$.
• Prove that under these assumptions, the total weighted volume $\int_M e^{-f} dv < \infty$.
• Apply the general result to self-shrinkers in $\mathbb{R}^{n+1}$ by setting $f = |x|^2/4$, leading to volume growth $\text{Vol}(B_r \cap \Sigma) \leq C r^{n - 2\beta}$.
• Use the equivalence of conditions (i)–(iv) in Theorem 1.3 to reframe the Colding-Minicozzi compactness theorem in terms of finite weighted volume.

Experimental results

Research questions

• RQ1 What is the precise volume growth rate of a complete noncompact self-shrinker in Euclidean space under the assumption of proper immersion?
• RQ2 How are the properties of properness, polynomial volume growth, and finite weighted volume related for self-shrinkers?
• RQ3 Can the volume growth of gradient shrinking Ricci solitons be estimated without curvature assumptions, using only the potential function?
• RQ4 Is the volume growth bound $\text{Vol}(B_r(x_0)) \leq C r^{n - 2\beta}$ sharp for gradient shrinking solitons?
• RQ5 Does the finiteness of the weighted volume $\int_{\Sigma} e^{-|x|^2/4} dv$ imply properness for self-shrinkers?

Key findings

• For a complete noncompact self-shrinker $\Sigma^n$ in $\mathbb{R}^{n+1}$ with $\beta = \inf H^2$, the volume satisfies $\text{Vol}(B_r(0) \cap \Sigma) \leq C r^{n - 2\beta}$ for $r \geq 1$, with $C > 0$ depending on $\Sigma$.
• The upper bound $r^{n - 2\beta}$ is optimal, as shown by examples like $\mathbb{R}^n$ and $S^k(\sqrt{2k}) \times \mathbb{R}^{n-k}$.
• For any complete immersed self-shrinker in $\mathbb{R}^{n+1}$, the following are equivalent: (i) properness, (ii) Euclidean volume growth, (iii) polynomial volume growth, and (iv) finite weighted volume $\int_{\Sigma} e^{-|x|^2/4} dv < \infty$.
• The volume growth estimate for gradient shrinking Ricci solitons is $\text{Vol}(B_r(x_0)) \leq C r^{n - 2\beta}$ for $r \geq 1$, where $\beta = \inf R$.
• The bound is sharp, with equality in the case of the flat Euclidean space and cylinder shrinking solitrons.
• The method provides an alternative proof of the Euclidean volume growth estimate for gradient shrinking solitons, distinct from Cao and Zhou (2009).