On the Morse Index with Constraints II: Applications

TL;DR

This paper applies an abstract Morse index framework to constrained optimization problems involving capillary surfaces, providing index estimates, classifications, and sharp inequalities for key examples like catenoids and cylinders.

Contribution

It extends Morse index theory to constrained capillary surface problems, offering new index estimates, classifications, and sharp inequalities for classical examples.

Findings

Index estimates recover known stability results.

Inequalities are shown to be sharp through examples.

Precise indices are determined for catenoids, cylinders, and CMC surfaces.

Abstract

This is the second paper in our sequence. Here, we apply our abstract Morse index formulation developed in the previous paper to study several optimization set-ups with constraints, including type I or/and type II considerations. A common theme is that critical points belong to the family of capillary surfaces, defined by constant mean curvature and intersecting the ambient manifold at a fixed angle. In each case, we classify how the general index is related to the index with a constraint. For capillary surfaces in a Euclidean ball, we obtain an index estimate which recovers stability results of G. Wang and C. Xia and J. Gou and C. Xia as special cases. By considering a family of examples, we show that inequality is also sharp. Furthermore, we precisely determine indices with constraints for important examples such as the critical catenoid, round cylinders in a ball, and CMC surfaces with constant curvature in a sphere.