Abstract
In this paper we study a boundary value problem in manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if we start with a metric of positive curvature and convex boundary (positive Second Fundamental form), the Ricci flow uniformizes the curvature. In the case of a metric with rotational symmetry, we indicate how to weaken the hypothesis to positive Ricci curvature and positive Second Fundamental form.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alexander S., Bishop R.: Comparison theorems for curves of bounded geodesic curvature in metric spaces with curvature bounded above. Differ. Geom. Appl. 6(1), 67–86 (1996)
Brendle S.: Curvature flows on surfaces with boundary. Math. Ann. 324(3), 491–519 (2002)
Chen X.-Z, Dong T.: Ricci deformation of a metric on a Riemannian manifold with boundary. J. Zhejiang Univ. Sci. Ed. 33(5), 496–499 (2006)
Chen B.-L., Zhu X.-P.: Complete Riemannian manifolds with pointwise pinched curvature. Invent. Math. 140, 423–452 (2000)
Cortissoz J.C.: The Ricci flow on the rotationally symmetric two-ball. Russian Math. (Iz VUZ). 51(12), 30–51 (2007)
Friedman A.: Partial differential equations of parabolic type. Prentice Hall, Inc., Englewood Cliffs, NJ (1964)
Hamilton R.S.: Three manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)
Hamilton R.S.: The formation of singularities in the Ricci flow. Surveys Differ. Geometry 2, 7–136 (1995)
Li, T. :The Ricci flow on surfaces with boundary. Ph.D. thesis, Universityof California at San Diego (1993)
Shen Y.: On Ricci deformation of a Riemannian metric on manifold with boundary. Pacific J. Math. 173(1), 203–221 (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cortissoz, J.C. Three-manifolds of positive curvature and convex weakly umbilic boundary. Geom Dedicata 138, 83–98 (2009). https://doi.org/10.1007/s10711-008-9300-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-008-9300-y