Mean value surfaces with prescribed curvature form
(2004) In Journal des Mathématiques Pures et Appliquées 83(9). p.10751107 Abstract
 The Gaussian curvature of a twodimensional 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 socalled metric potential, we are looking... (More)
 The Gaussian curvature of a twodimensional 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 socalled 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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/266387
 author
 Hedenmalm, Håkan ^{LU} and PerdomoGallipoli, Yolanda ^{LU}
 organization
 publishing date
 2004
 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
 00217824
 DOI
 10.1016/j.matpur.2004.03.001
 language
 English
 LU publication?
 yes
 id
 cbb99b01f01c445d8705918a669dd2b5 (old id 266387)
 date added to LUP
 20071026 15:50:53
 date last changed
 20171126 03:48:58
@article{cbb99b01f01c445d8705918a669dd2b5, abstract = {The Gaussian curvature of a twodimensional 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 socalled 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 PerdomoGallipoli, Yolanda}, issn = {00217824}, keyword = {mean value property,minimal area,bordered surface,Riemannian metric,curvature form}, language = {eng}, number = {9}, pages = {10751107}, 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}, }