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Minimal graphs over non-compact domains in 3-manifolds fibered by a Killing vector field

Andrea Del Prete

TL;DR

The work generalizes the Dirichlet problem for minimal graphs to 3-manifolds equipped with a Killing submersion, establishing Collin–Krüst type height-growth estimates and existence results for minimal Killing graphs over unbounded domains. It proves a higher-level uniqueness result in Nil$_3$ for bounded boundary data on strips and a removable singularities theorem for prescribed mean curvature, with methods that avoid relying on supersolutions. The framework is then extended to higher dimensions, where the mean curvature equation and barrier arguments adapt to the density-weighted setting, enabling analogous existence, uniqueness, and regularity results. Collectively, the paper broadens the landscape of minimal and prescribed-mean-curvature graphs in Killing submersions, linking domain geometry, bundle curvature, and vertical growth in a unified theory with potential applications to geometric analysis on homogeneous 3-manifolds.

Abstract

Let $\mathbb{E}$ be a connected and orientable Riemannian 3-manifold with a non-singular Killing vector field whose associated one-parameter group of the isometries of $\mathbb{E}$ acts freely and properly on $\E$. Then, there exists a Killing Submersion from $\E$ onto a connected and orientable surface $M$ whose fibers are the integral curves of the Killing vector field. In this setting, assuming that $M$ is non-compact and the fibers have infinite length, we solve the Dirichlet problem for minimal Killing graphs over certain unbounded domains of $M$, prescribing piecewise continuous boundary values. We obtain general Collin-Krust type estimates. In the particular case of the Heisenberg group, we prove a uniqueness result for minimal Killing graphs with bounded boundary values over a strip. We also prove that isolated singularities of Killing graphs with prescribed mean curvature are removable.

Minimal graphs over non-compact domains in 3-manifolds fibered by a Killing vector field

TL;DR

The work generalizes the Dirichlet problem for minimal graphs to 3-manifolds equipped with a Killing submersion, establishing Collin–Krüst type height-growth estimates and existence results for minimal Killing graphs over unbounded domains. It proves a higher-level uniqueness result in Nil for bounded boundary data on strips and a removable singularities theorem for prescribed mean curvature, with methods that avoid relying on supersolutions. The framework is then extended to higher dimensions, where the mean curvature equation and barrier arguments adapt to the density-weighted setting, enabling analogous existence, uniqueness, and regularity results. Collectively, the paper broadens the landscape of minimal and prescribed-mean-curvature graphs in Killing submersions, linking domain geometry, bundle curvature, and vertical growth in a unified theory with potential applications to geometric analysis on homogeneous 3-manifolds.

Abstract

Let be a connected and orientable Riemannian 3-manifold with a non-singular Killing vector field whose associated one-parameter group of the isometries of acts freely and properly on . Then, there exists a Killing Submersion from onto a connected and orientable surface whose fibers are the integral curves of the Killing vector field. In this setting, assuming that is non-compact and the fibers have infinite length, we solve the Dirichlet problem for minimal Killing graphs over certain unbounded domains of , prescribing piecewise continuous boundary values. We obtain general Collin-Krust type estimates. In the particular case of the Heisenberg group, we prove a uniqueness result for minimal Killing graphs with bounded boundary values over a strip. We also prove that isolated singularities of Killing graphs with prescribed mean curvature are removable.
Paper Structure (7 sections, 19 theorems, 86 equations, 7 figures)

This paper contains 7 sections, 19 theorems, 86 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation of Killing Submersions
  3. Existence of minimal Killing graphs over unbounded domains
  4. General Collin-Krust type results
  5. A uniqueness results in a strip of the Heisenberg group
  6. Removable singularities Theorem
  7. Generalizations to higher dimension

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^2$ be an unbounded domain and let $u,\tilde{u}\in C^2(\Omega)$ be such that $u_{\mid\partial\Omega}=\tilde{u}_{\mid\partial\Omega}$ and Denote $\Lambda(r)=\Omega\cap\left\{(x,y)\in\mathbb{R}^2: x^2+y^2=r\right\}$ and $M(r)=\underset{\Lambda(r)}{\sup} \,| u-\tilde{u}|$. Hence, Furthermore, if the length of $\Lambda(r)$ is uniformly bounded, then $\underset{r\to\infty}

Figures (7)

  • Figure 1: Sequence of domains in the $\mu$-wedge
  • Figure 2: Sequence of domains in the $\mu$-strip
  • Figure 3: A domain in $\mathbb{H}^2$ whose expansion rate function is $(r+1)\log(r+1)=f_1(r)$.
  • Figure 4: $(\theta_1,\theta_2)$-wedge.
  • Figure 5: Domain in $\mathbb{H}^2$ such that $\Lambda(\rho)$ is uniformly bounded.
  • ...and 2 more figures

Theorems & Definitions (38)

  • Theorem 1.1
  • Lemma 2.1: DMN
  • Lemma 2.2: DMN
  • Theorem 2.3
  • Theorem 2.4: DMN
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • proof
  • ...and 28 more