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The intrinsic metric of constant mean curvature surfaces and minimal hypersurfaces with free boundary

Lucas Ambrozio

TL;DR

The work revisits Ricci-Curbastro's intrinsic criteria for representing a surface metric as the first fundamental form of a (CMC) immersion into space forms and extends these ideas to constant mean curvature surfaces that meet umbilical boundaries orthogonally, including a Lawson-type free boundary correspondence. In dimension two, it shows that for an immersion with constant mean curvature $H$ into $\mathbb{Q}^3(c)$, the quantity $c+H^2-K$ is non-negative and the rescaled metric $\sqrt{c+H^2-K}\,g$ is flat away from the zero set of the traceless part $\mathring{A}$ of the second fundamental form, providing an intrinsic criterion for local immersion. The results generalize to higher dimensions via the dCaj framework, introducing an intrinsic tensor $\overline{g}=c(n-1)g-Ric_g$ and a factorisation with $A\circ A$, to obtain local minimal immersions into $\mathbb{Q}^{n+1}(c)$ with free boundary along umbilical boundaries; a corresponding boundary Lawson-type correspondence links spaces with the same $c+H^2$. The paper also develops intrinsic properties such as Ricci surfaces with boundary and connects these to Steklov eigenvalue problems for free boundary minimal surfaces, highlighting spectral aspects of the geometry. Overall, it provides a coherent intrinsic approach to free boundary CMC/minimal surfaces, their higher-dimensional generalisations, and related spectral phenomena.

Abstract

Ricci-Curbastro established necessary and sufficient conditions for a Riemannian metric on a surface to be the first fundamental form of a minimal immersion of that surface into the Euclidean space. We revisit certain developments arising from his theorem, and propose new versions of these results in the context of the theory of constant mean curvature surfaces in three-dimensional space forms that meet umbilical surfaces orthogonally along their boundary components. Higher dimensional generalisations, inspired by a theorem of do Carmo and Dajczer, are discussed as well.

The intrinsic metric of constant mean curvature surfaces and minimal hypersurfaces with free boundary

TL;DR

The work revisits Ricci-Curbastro's intrinsic criteria for representing a surface metric as the first fundamental form of a (CMC) immersion into space forms and extends these ideas to constant mean curvature surfaces that meet umbilical boundaries orthogonally, including a Lawson-type free boundary correspondence. In dimension two, it shows that for an immersion with constant mean curvature into , the quantity is non-negative and the rescaled metric is flat away from the zero set of the traceless part of the second fundamental form, providing an intrinsic criterion for local immersion. The results generalize to higher dimensions via the dCaj framework, introducing an intrinsic tensor and a factorisation with , to obtain local minimal immersions into with free boundary along umbilical boundaries; a corresponding boundary Lawson-type correspondence links spaces with the same . The paper also develops intrinsic properties such as Ricci surfaces with boundary and connects these to Steklov eigenvalue problems for free boundary minimal surfaces, highlighting spectral aspects of the geometry. Overall, it provides a coherent intrinsic approach to free boundary CMC/minimal surfaces, their higher-dimensional generalisations, and related spectral phenomena.

Abstract

Ricci-Curbastro established necessary and sufficient conditions for a Riemannian metric on a surface to be the first fundamental form of a minimal immersion of that surface into the Euclidean space. We revisit certain developments arising from his theorem, and propose new versions of these results in the context of the theory of constant mean curvature surfaces in three-dimensional space forms that meet umbilical surfaces orthogonally along their boundary components. Higher dimensional generalisations, inspired by a theorem of do Carmo and Dajczer, are discussed as well.
Paper Structure (13 sections, 4 theorems, 55 equations)

This paper contains 13 sections, 4 theorems, 55 equations.

Table of Contents

  1. The first fundamental form of CMC immersions
  2. Motivation
  3. Dimension two
  4. The Lawson correspondence
  5. Higher dimensions
  6. The free boundary condition
  7. Umbilical hypersurfaces and capillary boundary conditions
  8. Dimension two
  9. The free boundary correspondence
  10. Higher dimensions
  11. Other intrinsic properties
  12. Ricci surfaces with boundary
  13. Steklov eigenvalues

Key Result

Lemma 2.1.1

Under the capillary conditions, for every tangent vector fields $X$ and $Y$ on $\partial_i \Sigma$, we have In particular, in the free boundary case, where the sign of the trace of $B_i$ depends on whether $\nu=\overline{N}$ or $\nu=-\overline{N}$.

Theorems & Definitions (17)

  • proof
  • Remark 1.2.1
  • Example 1.3.1
  • proof
  • Remark 1.4.1
  • Lemma 2.1.1
  • proof
  • Lemma 2.1.2
  • proof
  • proof
  • ...and 7 more