Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Prescribing curvatures on three dimensional Riemannian manifolds with boundaries

Author: Lei Zhang
Journal: Trans. Amer. Math. Soc. 361 (2009), 3463-3481
MSC (2000): Primary 35J60, 53B20
Published electronically: February 23, 2009
MathSciNet review: 2491888
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (M,g)$ be a complete three dimensional Riemannian manifold with boundary $ \partial M$. Given smooth functions $ K(x)>0$ and $ c(x)$ defined on $ M$ and $ \partial M$, respectively, it is natural to ask whether there exist metrics conformal to $ g$ so that under these new metrics, $ K$ is the scalar curvature and $ c$ is the boundary mean curvature. All such metrics can be described by a prescribing curvature equation with a boundary condition. With suitable assumptions on $ K$,$ c$ and $ (M,g)$ we show that all the solutions of the equation can only blow up at finite points over each compact subset of $ \bar M$; some of them may appear on $ \partial M$. We describe the asymptotic behavior of the blow-up solutions around each blow-up point and derive an energy estimate as a consequence.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J60, 53B20

Retrieve articles in all journals with MSC (2000): 35J60, 53B20

Additional Information

Lei Zhang
Affiliation: Department of Mathematics, University of Alabama at Birmingham, 1300 University Boulevard, 452 Campbell Hall, Birmingham, Alabama 35294-1170

PII: S 0002-9947(09)04911-3
Keywords: Scalar curvature, mean curvature, Harnack inequality
Received by editor(s): September 13, 2006
Published electronically: February 23, 2009
Additional Notes: The author was supported by National Science Foundation Grant 0600275 (0810902)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia