Good shadows, dynamics, and convex hulls

TL;DR

This paper explores the existence of good shadows in variational principles, providing new proofs and extending results using dynamical systems and volume ratio techniques, with applications to convex hulls of submanifolds.

Contribution

It offers a new dynamical systems proof of the strong version of the Yau minimum principle and establishes a special case of a conjecture on good shadows in the $C^2$ setting.

Findings

Proof of the strong Yau minimum principle using dynamical systems.

Establishment of a special case of the good shadows conjecture.

Application to convex hulls of submanifolds with controlled Gauss maps.

Abstract

The Ekeland variational principle implies what can be regarded as a strong version, in the $C^1$ category, of the Yau minimum principle: under the appropriate hypotheses {\it every} minimizing sequence admits a {\it good shadow}, a second minimizing sequence that has good properties and is asymptotic to the original one. Using arguments from dynamical systems, we give another proof of this result and also establish, with the aid of Gromov's theorem on monotonicity of volume ratios, a special case of a conjecture claiming the existence of good shadows in the original $C^2$ setting of the Yau minimum principle. The interest in having an abundance of good shadows stems from the fact that this is a desirable property if one wants to refine the applications of the asymptotic minimum principle, as it allows for information to be localized at infinity. These ideas are applied in this paper to the study of the convex hulls of complete submanifolds of Euclidean $n$-space that have controlled Grassmanian-valued Gauss maps.