Cohomology vanishing theorems for free boundary submanifolds

TL;DR

This paper proves cohomology vanishing theorems for free boundary submanifolds in Euclidean balls using a new Hardy inequality, addressing open questions on normal bundle flatness and pinching constants.

Contribution

It introduces a new Hardy inequality to establish cohomology vanishing theorems, removing the flatness condition and partially resolving open questions on pinching constants.

Findings

Established cohomology vanishing theorems under various pinching conditions.

Removed the normal bundle flatness condition from previous results.

Provided partial answers to open questions on optimal pinching constants.

Abstract

In this paper, via a new Hardy type inequality, we establish some cohomology vanishing theorems for free boundary compact submanifolds $M^n$ with $n\geq2$ immersed in the Euclidean unit ball $\mathbb{B}^{n+k}$ under one of the pinching conditions $|\Phi|^2\leq C$, $|A|^2\leq \widetilde{C}$, or $|\Phi|\leq R(p,|H|)$, where $A$ $(\Phi)$ is the (traceless) second fundamental form, $H$ is the mean curvature, $C,\widetilde{C}$ are positive constants and $R(p,|H|)$ is a positive function. In particular, we remove the condition on the flatness of the normal bundle, solving the first question, and partially answer the second question on optimal pinching constants proposed by Cavalcante, Mendes and Vit\'{o}rio.