AN ISOPERIMETRIC EQUALITY, AND RELATED QUESTIONS

Abstract

Let S be a piece of smooth two-dimensional surface. If circles of radius r on the surface are defined with respect to global conformal parameters on S, then an estimate of the ratio of the square of the length of the circle to the area of the circle is shown to be true for every complete simply-connected open surface on which the curvature is absolutely integrable. If certain smoothness hypotheses are satisfied by S at infinity, then these circles become, up to terms of negligible order, curves which are equidistant from a fixed point on S. In this sense, the image of a large circle in a plane on which S is represented conformally will be a circle on S. As corollaries of the method, asymptotic estimates for the lengths of the images of circles and of radial lines in such a reference plane are given, depending only on the curvatura integra.

Open PDF

Document Details

Document Type
Technical Report
Publication Date
Nov 14, 1961
Accession Number
AD0268213

Entities

People

  • Robert Finn

Organizations

  • Stanford University

Tags

Communities of Interest

  • Air Platforms

DTIC Thesaurus Topics

  • Aeronautical Engineering
  • Analytic Functions
  • Applied Mathematics
  • Complex Variables
  • Contracts
  • Curvature
  • Engineering
  • Geometry
  • Government Procurement
  • Mathematics
  • Military Research
  • Navy
  • New York
  • Ordnance Laboratories
  • Rhode Island
  • Two Dimensional
  • United States

Fields of Study

  • Mathematics

Readers

  • Approximation Theory.
  • Graph Algorithms and Convex Optimization.