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Half-space type theorems for a class of weighted minimal surfaces in $\mathbb{R}^{3}$

A. L. Martínez-Triviño, J. P. dos Santos, G. Tinaglia

TL;DR

The paper investigates half-space phenomena for a class of height-dependent weighted minimal surfaces in $\mathbb{R}^3$, defined as critical points of the weighted area $\mathcal{A}^{\varphi}$ with $\varphi$ depending only on height. By employing the weak maximum principle and the Omori–Yau maximum principle for the drift Laplacian $\Delta^{\varphi}$ on $\varphi$-stochastically complete surfaces, it establishes a suite of half-space and wedge nonexistence results (Theorems A–D) and a nonexistence result in a right circular cone (Theorem E), together with a strong half-space type theorem. The approach hinges on constructing barrier functions, computing $\Delta^{\varphi}$ of height- or distance-type functions, and deriving contradictions under appropriate growth and completeness hypotheses. The results extend classical half-space theorems to the weighted setting and connect to translator and $\alpha$-minimal surface theory, with implications for the global geometry and rigidity of weighted minimal surfaces in $\mathbb{R}^3$.

Abstract

We establish half-space type results for a class of height-dependent weighted minimal surfaces in $\mathbb{R}^3$, namely critical points of a weighted area functional whose weight depends on the height. When the weight has at most quadratic growth, we prove that there are no proper surfaces contained either in two transverse vertical half-spaces of $\mathbb{R}^3$ or in a half-space determined by a non-vertical plane. We show that this second result holds in a more general context, namely, for a class of stochastically complete weighted minimal surfaces. In this setup, we also prove a result for surfaces contained in regions bounded by cones. Furthermore, for stochastically complete weighted minimal surfaces satisfying restrictions on their principal curvatures, we establish a version of the classic strong half-space result due to Hoffman-Meeks.

Half-space type theorems for a class of weighted minimal surfaces in $\mathbb{R}^{3}$

TL;DR

The paper investigates half-space phenomena for a class of height-dependent weighted minimal surfaces in , defined as critical points of the weighted area with depending only on height. By employing the weak maximum principle and the Omori–Yau maximum principle for the drift Laplacian on -stochastically complete surfaces, it establishes a suite of half-space and wedge nonexistence results (Theorems A–D) and a nonexistence result in a right circular cone (Theorem E), together with a strong half-space type theorem. The approach hinges on constructing barrier functions, computing of height- or distance-type functions, and deriving contradictions under appropriate growth and completeness hypotheses. The results extend classical half-space theorems to the weighted setting and connect to translator and -minimal surface theory, with implications for the global geometry and rigidity of weighted minimal surfaces in .

Abstract

We establish half-space type results for a class of height-dependent weighted minimal surfaces in , namely critical points of a weighted area functional whose weight depends on the height. When the weight has at most quadratic growth, we prove that there are no proper surfaces contained either in two transverse vertical half-spaces of or in a half-space determined by a non-vertical plane. We show that this second result holds in a more general context, namely, for a class of stochastically complete weighted minimal surfaces. In this setup, we also prove a result for surfaces contained in regions bounded by cones. Furthermore, for stochastically complete weighted minimal surfaces satisfying restrictions on their principal curvatures, we establish a version of the classic strong half-space result due to Hoffman-Meeks.
Paper Structure (8 sections, 65 equations, 3 figures)

This paper contains 8 sections, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Brief introduction to weak maximum principle
  3. Proofs of Theorems A and C
  4. Proof of Theorem B
  5. Proof of Theorem D
  6. Proof of Theorem E
  7. Concluding remarks
  8. Acknowledgments

Figures (3)

  • Figure 1.1: A wedge region in $\mathbb{R}^3$ (left). A complete $[\varphi,\vec{e}_{3}]$-bowl contained in a vertical cylinder (right).
  • Figure 1.2: Region bounded by a cone
  • Figure 1.3: The grim reaper cylinder and its asymptotic behaviour between two planes

Theorems & Definitions (7)

  • proof
  • proof : Proof of Theorem C
  • proof : Proof of Theorem A
  • proof
  • proof
  • proof
  • proof