zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Ricci flow on Kähler-Einstein manifolds. (English) Zbl 1097.53045

There is a long-standing problem in Ricci flow: on a compact Kähler-Einstein manifold, does the Kähler-Ricci flow converge to a Kähler-Einstein metric if the initial metric has positive bisectional curvature? In this article authors give a complete affirmative answer to this problem. Theorem 1.1: Let $M$ be a Kähler-Einstein manifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive bisectional curvature at least at one point, then the Kähler-Ricci flow converges exponentially fast to a Kähler-Einstein metric with constant bisectional curvature. This problem was completely solved by R. S. Hamilton in the case of Riemann surfaces.

As a direct consequence, the authors have the following Corollary 1.3: The space of Kähler metrics with nonnegative bisectional curvature (and positive at least at one point) is path connected. The space of metrics with a nonnegative curvature operator (and positive at least at one point) is also path connected.

Theorem 1.4. Let $M$ be any Kähler-Einstein orbifold with positive scalar curvature. If the initial metric has nonnegative bisectional curvature and positive bisectional curvature at least at one point, then the Kähler-Ricci flow converges exponentially fast to a Kähler-Einstein metric with constant bisectional curvature. Moreover, $M$ is a global quotient of $ℂ{P}^{n}·$

MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 32Q20 Kähler-Einstein manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C55 Hermitian and Kählerian manifolds (global differential geometry)