Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Surfaces and Hypersurfaces with Prescribed Radial Mean Curvature

Marcelo Lopes Ferro, Armando M. V. Corro

TL;DR

The paper addresses the problem of classifying hypersurfaces in Euclidean spaces with prescribed radial mean curvature by introducing rotated translational hypersurfaces and leveraging height and angle functions. It develops a general framework where the radial mean curvature is encoded by the equation $N_1 A_X = a X_1 + b$ with $A_X = n H_R$, and proves key results including a broad structural classification and a recursive construction method for higher dimensions. It provides explicit local classifications for CPD, translational, and harmonic graph-type surfaces under this curvature prescription, plus a complete local 3D $H_{n-1}=0$ classification and a parallel-hypersurface toolkit to generate further examples. The work offers a systematic approach to constructing curvature-prescribed hypersurfaces across dimensions, enriching the catalog of CPD and graph-type surfaces and enabling recursive generation of new higher-dimensional examples.

Abstract

In this work, we provide a local classification of certain special classes of surfaces determined by the prescription of the radial mean curvature in terms of the height and angle functions. Moreover, we introduce a special class of hypersurfaces, and we also provide a local classification of these three-dimensional hypersurfaces whose second mean curvature vanishes. Finally, we present a recursive method for constructing such hypersurfaces, extending the same curvature prescription approach to higher dimensions.

Surfaces and Hypersurfaces with Prescribed Radial Mean Curvature

TL;DR

The paper addresses the problem of classifying hypersurfaces in Euclidean spaces with prescribed radial mean curvature by introducing rotated translational hypersurfaces and leveraging height and angle functions. It develops a general framework where the radial mean curvature is encoded by the equation with , and proves key results including a broad structural classification and a recursive construction method for higher dimensions. It provides explicit local classifications for CPD, translational, and harmonic graph-type surfaces under this curvature prescription, plus a complete local 3D classification and a parallel-hypersurface toolkit to generate further examples. The work offers a systematic approach to constructing curvature-prescribed hypersurfaces across dimensions, enriching the catalog of CPD and graph-type surfaces and enabling recursive generation of new higher-dimensional examples.

Abstract

In this work, we provide a local classification of certain special classes of surfaces determined by the prescription of the radial mean curvature in terms of the height and angle functions. Moreover, we introduce a special class of hypersurfaces, and we also provide a local classification of these three-dimensional hypersurfaces whose second mean curvature vanishes. Finally, we present a recursive method for constructing such hypersurfaces, extending the same curvature prescription approach to higher dimensions.

Paper Structure

This paper contains 6 sections, 13 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and statement of the main results
  3. Auxiliary results on ordinary differential equations
  4. CPD, Translational, and Harmonic Surfaces with radial mean curvature prescribed in terms of the height and angle functions
  5. Technical results on parallel hypersurfaces
  6. Rotated translational hypersurfaces

Key Result

Theorem 2.1

Let $M^n$ be a rotated translational hypersurface of dimension $n \geq 3$, whose profile is a curve. Then, the $(n-1)$-mean curvature of $M^n$ is zero, that is, $H_{n-1}=0$, if and only if, there exist constants $C_1 > 0$, $C_2 \in \mathbb{R}$, such that the directrix of $M^n$ is either a hypersurfa where $\theta \in \left(-\tfrac{\pi}{2},\,\tfrac{\pi}{2}\right)$.

Theorems & Definitions (44)

  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2.1
  • Theorem 2.2
  • Remark 4
  • Lemma 1
  • proof
  • Remark 5
  • ...and 34 more