[Paper Review] Geometric inequalities for static convex domains in hyperbolic space

Yingxiang Hu, Haizhong Li|arXiv (Cornell University)|May 9, 2021

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper establishes new geometric inequalities for static convex hypersurfaces in hyperbolic space by analyzing two classes of locally constrained curvature flows. It proves that static convexity is preserved under these flows, and using long-time convergence to geodesic spheres, derives a family of quermassintegral inequalities that generalize previous results to a broader class of domains beyond h-convex or star-shaped ones.

ABSTRACT

We prove that the static convexity is preserved along two kinds of locally constrained curvature flows in hyperbolic space. Using the static convexity of the flow hypersurfaces, we prove new family of geometric inequalities for such hypersurfaces in hyperbolic space.

Motivation & Objective

• To extend quermassintegral inequalities in hyperbolic space beyond h-convex or star-shaped domains.
• To establish the preservation of static convexity under two classes of locally constrained curvature flows in hyperbolic space.
• To prove long-time existence and exponential convergence of the flows to geodesic spheres.
• To derive new geometric inequalities involving quermassintegrals using the convergence properties of the flows.
• To generalize known inequalities to domains with nonnegative sectional curvature and static convexity.

Proposed method

• Introduces two locally constrained curvature flows: one with speed $ F = E_k/E_{k-1} $ and another with $ F = E_1 $, both designed to preserve certain quermassintegrals.
• Uses the Minkowski formula in hyperbolic space to show that the flows preserve $ W_k $ and control the evolution of $ W_m $ and $ W_0 $.
• Applies $ C^0 $-estimates and curvature estimates to establish long-time existence of the flows.
• Proves that static convexity is preserved under the flows by analyzing the evolution of the support function and principal curvatures.
• Employs maximum principle arguments and the positivity of the Hessian of the support function to control the evolution of geometric quantities.
• Uses convergence to geodesic spheres as $ t \to \infty $ to derive inequalities between quermassintegrals via comparison with geodesic balls.

Experimental results

Research questions

• RQ1 Can quermassintegral inequalities in hyperbolic space be extended to domains that are not h-convex or star-shaped?
• RQ2 Does static convexity persist under locally constrained curvature flows in hyperbolic space?
• RQ3 Can the convergence of such flows to geodesic spheres be established under weaker convexity assumptions?
• RQ4 What geometric inequalities can be derived from the long-time behavior of these curvature flows?
• RQ5 Under what conditions does equality in the geometric inequalities occur?

Key findings

• The static convexity of hypersurfaces is preserved along the locally constrained curvature flows $ \partial_t X = (E_k/E_{k-1} - u/\lambda') \nu $ and $ \partial_t X = (E_1 - u/\lambda') \nu $ in hyperbolic space.
• The flows exist for all time and converge smoothly to geodesic spheres centered at the origin.
• For $ 1 \leq m \leq k \leq n $, the inequality $ W_{k+1}^{\lambda'}(\Omega) \geq h_{k+1} \circ f_m^{-1}(W_m(\Omega)) $ holds for static convex domains, with equality if and only if $ \Omega $ is a geodesic ball.
• For $ k=0 $, the inequality $ W_1^{\lambda'}(\Omega) \geq h_1 \circ f_0^{-1}(W_0(\Omega)) $ holds, with equality only for geodesic balls.
• The results generalize previous inequalities from h-convex or star-shaped domains to the broader class of static convex domains with nonnegative sectional curvature.
• The convergence of the flows to geodesic spheres implies that the quermassintegral inequalities are sharp and equality holds precisely for geodesic balls.