The Bj\"orling problem for prescribed mean curvature surfaces in $\mathbb{R}^3$

TL;DR

This paper solves the Björling problem for surfaces in three-dimensional space with prescribed mean curvature depending on the Gauss map, enabling the construction of new surfaces including M"obius strip topologies and self-translating solitons.

Contribution

It introduces a method to solve the Björling problem for prescribed mean curvature surfaces and constructs novel examples with complex topologies.

Findings

Existence of surfaces with M"obius strip topology for various prescribed functions.

Construction of the first known self-translating solitons with M"obius topology.

Extension of Björling problem solutions to prescribed mean curvature surfaces.

Abstract

In this paper we solve the Bj\"orling problem for the class of immersed surfaces in $\mathbb{R}^3$ whose mean curvature is given as an analytic function depending on its Gauss map. As an application, we prove the existence of surfaces with the topology of a M\"obius strip for an arbitrary large class of prescribed functions. In particular, we use the Bj\"orling problem to construct the first known examples of self-translating solitons of the mean curvature flow with the topology of a M\"obius strip in $\mathbb{R}^3$