Operations on Legendrian submanifolds

TL;DR

This paper introduces sum and convolution operations on Legendrian submanifolds in the 1-jet space, linking them to convex analysis and refining the generating function theory with min-max selectors.

Contribution

It defines new operations on Legendrian submanifolds that extend classical convex analysis techniques and integrates min-max selectors into generating function theory.

Findings

Defined sum and convolution operations for Legendrian submanifolds.

Linked these operations to convex analysis functions.

Refined generating function theory with min-max selectors.

Abstract

We focus on Legendrian submanifolds of the space of one-jets of functions, $J^1(\mathbb{R}^n,\mathbb{R})$. We are interested in processes - operations - that build new Legendrian submanifolds from old ones. We introduce in particular two operations, namely the sum and the convolution, which in some sense lift to $J^1(\mathbb{R}^n,\mathbb{R})$ the operations sum and infimal-convolution on functions that belong to convex analysis. We show that these operations fit well with the classical theory of generating functions. Finally, we refine this theory so that the min-max selector of generating functions plays its natural role.