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Mean value surfaces with prescribed curvature form

Hedenmalm, Håkan LU and Perdomo-Gallipoli, Yolanda LU (2004) In Journal des Mathématiques Pures et Appliquées 83(9). p.1075-1107
Abstract
The Gaussian curvature of a two-dimensional Riemannian manifold is uniquely determined by the choice of the metric. The formulas for computing the curvature in terms of components of the metric, in isothermal coordinates, involve the Laplacian operator and therefore, the problem of finding a Riemannian metric for a given curvature form may be viewed as a potential theory problem. This problem has, generally speaking, a multitude of solutions. To specify the solution uniquely, we ask that the metric have the mean value property for harmonic functions with respect to some given point. This means that we assume that the surface is simply connected and that it has a smooth boundary. In terms of the so-called metric potential, we are looking... (More)
The Gaussian curvature of a two-dimensional Riemannian manifold is uniquely determined by the choice of the metric. The formulas for computing the curvature in terms of components of the metric, in isothermal coordinates, involve the Laplacian operator and therefore, the problem of finding a Riemannian metric for a given curvature form may be viewed as a potential theory problem. This problem has, generally speaking, a multitude of solutions. To specify the solution uniquely, we ask that the metric have the mean value property for harmonic functions with respect to some given point. This means that we assume that the surface is simply connected and that it has a smooth boundary. In terms of the so-called metric potential, we are looking for a unique smooth solution to a nonlinear fourth order elliptic partial differential equation with second order Cauchy data given on the boundary. We find a simple condition on the curvature form which ensures that there exists a smooth mean value surface solution. It reads: the curvature form plus half the curvature form for the hyperbolic plane (with the same coordinates) should be less than or equal to 0. The same analysis leads to results on the question of whether the canonical divisors in weighted Bergman spaces over the unit disk have extraneous zeros. Numerical work suggests that the above condition on the curvature form is essentially sharp. Our problem is in spirit analogous to the classical Minkowski problem, where the sphere supplies the chart coordinates via the Gauss map. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
mean value property, minimal area, bordered surface, Riemannian metric, curvature form
in
Journal des Mathématiques Pures et Appliquées
volume
83
issue
9
pages
1075 - 1107
publisher
Elsevier
external identifiers
  • wos:000224008300001
  • scopus:4444305803
ISSN
0021-7824
DOI
10.1016/j.matpur.2004.03.001
language
English
LU publication?
yes
id
cbb99b01-f01c-445d-8705-918a669dd2b5 (old id 266387)
date added to LUP
2007-10-26 15:50:53
date last changed
2017-11-26 03:48:58
@article{cbb99b01-f01c-445d-8705-918a669dd2b5,
  abstract     = {The Gaussian curvature of a two-dimensional Riemannian manifold is uniquely determined by the choice of the metric. The formulas for computing the curvature in terms of components of the metric, in isothermal coordinates, involve the Laplacian operator and therefore, the problem of finding a Riemannian metric for a given curvature form may be viewed as a potential theory problem. This problem has, generally speaking, a multitude of solutions. To specify the solution uniquely, we ask that the metric have the mean value property for harmonic functions with respect to some given point. This means that we assume that the surface is simply connected and that it has a smooth boundary. In terms of the so-called metric potential, we are looking for a unique smooth solution to a nonlinear fourth order elliptic partial differential equation with second order Cauchy data given on the boundary. We find a simple condition on the curvature form which ensures that there exists a smooth mean value surface solution. It reads: the curvature form plus half the curvature form for the hyperbolic plane (with the same coordinates) should be less than or equal to 0. The same analysis leads to results on the question of whether the canonical divisors in weighted Bergman spaces over the unit disk have extraneous zeros. Numerical work suggests that the above condition on the curvature form is essentially sharp. Our problem is in spirit analogous to the classical Minkowski problem, where the sphere supplies the chart coordinates via the Gauss map.},
  author       = {Hedenmalm, Håkan and Perdomo-Gallipoli, Yolanda},
  issn         = {0021-7824},
  keyword      = {mean value property,minimal area,bordered surface,Riemannian metric,curvature form},
  language     = {eng},
  number       = {9},
  pages        = {1075--1107},
  publisher    = {Elsevier},
  series       = {Journal des Mathématiques Pures et Appliquées},
  title        = {Mean value surfaces with prescribed curvature form},
  url          = {http://dx.doi.org/10.1016/j.matpur.2004.03.001},
  volume       = {83},
  year         = {2004},
}