On the Kähler-ricci Flow on Complex Surfaces
نویسنده
چکیده
One of the most important properties of a geometric flow is whether it preserves the positivity of various notions of curvature. In the case of the Kähler-Ricci flow, the positivity of the curvature operator (Hamilton [7]), the positivity of the biholomorphic sectional curvature (Bando [1], Mok[8]), and the positivity of the scalar curvature (Hamilton [4]) are all preserved. However, whether the positivity of the Ricci curvature is preserved is still not known. As stressed for example in Chen-Tian [3], this is central to the problem of convergence of the Kähler-Ricci flow on Kähler-Einstein manifolds of positive curvature. The existence of Kähler-Einstein metrics has been conjectured by S.T.Yau [10] to be equivalent to stability in geometric invariant theory, and there is strong interest in relating these notions to the behavior of the Kähler-Ricci flow.
منابع مشابه
A Property of Kähler-Ricci Solitons on Complete Complex Surfaces
where Rαβ(x, t) denotes the Ricci curvature tensor of the metric gαβ(x, t). One of the main problems in differential geometry is to find canonical structure on manifolds. The Ricci flow introduced by Hamilton [8] is an useful tool to approach such problems. For examples, Hamilton [10] and Chow [7] used the convergence of the Ricci flow to characterize the complex structures on compact Riemann s...
متن کاملGEOMETRIZATION OF HEAT FLOW ON VOLUMETRICALLY ISOTHERMAL MANIFOLDS VIA THE RICCI FLOW
The present article serves the purpose of pursuing Geometrization of heat flow on volumetrically isothermal manifold by means of RF approach. In this article, we have analyzed the evolution of heat equation in a 3-dimensional smooth isothermal manifold bearing characteristics of Riemannian manifold and fundamental properties of thermodynamic systems. By making use of the notions of various curva...
متن کاملConvergence of the Kähler-ricci Flow and Multiplier Ideal Sheaves on Del Pezzo Surfaces
On certain del Pezzo surfaces with large automorphism groups, it is shown that the solution to the Kähler-Ricci flow with a certain initial value converges in C∞-norm exponentially fast to a Kähler-Einstein metric. The proof is based on the method of multiplier ideal sheaves.
متن کاملThe Kähler Ricci Flow on Fano Surfaces (I)
Suppose {(M, g(t)), 0 ≤ t <∞} is a Kähler Ricci flow solution on a Fano surface. If |Rm| is not uniformly bounded along this flow, we can blowup at the maximal curvature points to obtain a limit complete Riemannian manifold X. We show that X must have certain topological and geometric properties. Using these properties, we are able to prove that |Rm| is uniformly bounded along every Kähler Ricc...
متن کاملThe Kähler-ricci Flowon Kähler Surfaces
The problem of finding Kähler-Einstein metrics on a compact Kähler manifold has been the subject of intense study over the last few decades. In his solution to Calabi’s conjecture, Yau [Ya1] proved the existence of a Kähler-Einstein metric on compact Kähler manifolds with vanishing or negative first Chern class. An alternative proof of Yau’s theorem is given by Cao [Ca] using the Kähler-Ricci f...
متن کامل