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Families of surfaces with constant ratio of principal curvatures and Plateau's problem

Mikhail Skopenkov, Khusrav Yorov

TL;DR

The paper develops a framework for surfaces with a constant ratio of principal curvatures (CRPC) by coupling Euclidean and isotropic viewpoints. It proves that any analytic minimal surface without flat points sits inside a unique analytic family of CRPC surfaces with fixed boundary, yielding a partial Plateau-type solution; this is mirrored in isotropic geometry, where CRPC surfaces satisfy $2H=t\sqrt{-K}$. The authors advance the theory through an isotropic-to-Euclidean strategy, employing successive approximations and analytic majorization to establish real-analytic families and Schauder-type estimates, and they provide closed-form and second-order approximate CRPC surfaces to guide practical constructions such as gridshells. The work thus links classical minimal-surface theory with CRPC generalizations, offering both rigorous existence/analyticity results and practical approximations for design applications.

Abstract

This work is on surfaces with a constant ratio of principal curvatures. These CRPC surfaces generalize minimal surfaces but are much more challenging to construct. We propose a construction of a family of such surfaces containing a given minimal surface without flat points. This leads to a partial solution of Plateau's problem for CRPC surfaces. We obtain analogous results in isotropic geometry. This work illustrates a general approach to solving Euclidean problems by starting with their isotropic analogs. Besides, we apply the method of successive approximations and analytic majorization.

Families of surfaces with constant ratio of principal curvatures and Plateau's problem

TL;DR

The paper develops a framework for surfaces with a constant ratio of principal curvatures (CRPC) by coupling Euclidean and isotropic viewpoints. It proves that any analytic minimal surface without flat points sits inside a unique analytic family of CRPC surfaces with fixed boundary, yielding a partial Plateau-type solution; this is mirrored in isotropic geometry, where CRPC surfaces satisfy . The authors advance the theory through an isotropic-to-Euclidean strategy, employing successive approximations and analytic majorization to establish real-analytic families and Schauder-type estimates, and they provide closed-form and second-order approximate CRPC surfaces to guide practical constructions such as gridshells. The work thus links classical minimal-surface theory with CRPC generalizations, offering both rigorous existence/analyticity results and practical approximations for design applications.

Abstract

This work is on surfaces with a constant ratio of principal curvatures. These CRPC surfaces generalize minimal surfaces but are much more challenging to construct. We propose a construction of a family of such surfaces containing a given minimal surface without flat points. This leads to a partial solution of Plateau's problem for CRPC surfaces. We obtain analogous results in isotropic geometry. This work illustrates a general approach to solving Euclidean problems by starting with their isotropic analogs. Besides, we apply the method of successive approximations and analytic majorization.
Paper Structure (13 sections, 25 theorems, 74 equations, 4 figures)

This paper contains 13 sections, 25 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Main idea
  4. Previous work
  5. Organization of the paper
  6. Closed-form examples
  7. Euclidean CRPC surfaces
  8. Isotropic Geometry
  9. Isotropic CRPC surfaces
  10. Isotropic CRPC surfaces
  11. Euclidean CRPC surfaces
  12. Approximate CRPC surfaces
  13. Backup: Classical Results

Key Result

Theorem 1.1

Each minimal surface $\Phi^0\subset \mathbb{R}^3$ with analytic boundary and no flat points (even on $\partial\Phi^0$), which is the graph of a real analytic function in a closed Jordan domain, is contained in a unique (up to restriction) analytic family of surfaces $\Phi^s$ with the ratio $s-1$ of

Figures (4)

  • Figure 1: Left: A surface with a constant ratio of principal curvatures spanning a smooth curve. The asymptotic curves (dark) intersect at a constant angle ($80^\circ$). Right: An asymptotic gridshell obtained by bending originally rectangular strips and placing them orthogonal to a CRPC surface. The strips follow the asymptotic curves and intersect each other at a constant angle ($60^\circ$) except for a flat point in the middle. The gridshell was computed by optimizing an approximate CPRC surface.
  • Figure 2: An asymptotic gridshell by E. Schling schling:2018. The strips follow the asymptotic lines of a minimal surface and therefore intersect each other at a right angle.
  • Figure 3: Isotropic CRPC surfaces (from left to right): a hyperbolic paraboloid (Example \ref{['ex:eq-paraboloid']}), a rotational surface (Example \ref{['ex:rot-isotropic']}), a ruled surface (Example \ref{['ex:half-helicoid']}), a helical surface (Example \ref{['ex:part-helicatenoid']}), a translational surface (Example \ref{['ex:translational']}).
  • Figure 4: An approximate solution to Plateau's problem for CPRC surfaces. The asymptotic lines of the surface (gray) intersect at a constant angle ($80^\circ$). Fitting the prescribed boundary curve (green) is not perfect because our optimization algorithm gives higher preference to achieving the prescribed intersection angle (and positions of flat points if present).

Theorems & Definitions (57)

  • Theorem 1.1
  • Corollary 1.2: Partial solution to Plateau's problem for CRPC surfaces
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • ...and 47 more