Ricci flow on surfaces with conic singularities

Abstract

We establish the short-time existence of the Ricci flow on surfaces with a finite number of conic points, all with cone angle between 0 and $2\pi$, where the cone angles remain fixed or change in some smooth prescribed way. For the angle-preserving flow we prove long-time existence and convergence. When the Troyanov angle condition is satisfied (equivalently, when the data is logarithmically K-stable), the flow converges to the unique constant curvature metric with the given cone angles; if this condition is not satisfied, the flow converges subsequentially to a soliton. This is the one-dimensional version of the Hamilton--Tian conjecture.

v1: 38 pages v2: 39 pages, restructured Sections 1 and 2, and added references and Subsection 5.4. v3-v4: 41 pages, revised to address referee comments; original proof of Proposition 5.3 had an error pointed out to us by a referee. We fix this by invoking Chow and Hamilton's original arguments instead of the Hamilton compactness theorem. Final version. To appear in Analysis and PDE

References in corpus (9)

• Some discretizations of geometric evolution equations and the Ricci iteration on the space of Kahler metrics, I
• A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem
• On the construction of Nadel multiplier ideal sheaves and the limiting behavior of the Ricci flow
• Bessel Functions, Heat Kernel and the Conical Kähler-Ricci Flow
• The Ricci flow on the sphere with marked points
• Ricci Flow in Two Dimensions
• Gradient Ricci solitons on surfaces
• Regularity of Kähler-Ricci flow
• Polyhomogénéité des métriques asymptotiquement hyperboliques complexes le long du flot de Ricci

Cited by in corpus (22)

• Smooth approximations of the Conical Kahler-Ricci flows
• The conical Kähler-Ricci flow on Fano manifolds
• On the long time behaviour of the Conical Kähler- Ricci flows
• The Ricci flow on the sphere with marked points
• Smooth and singular Kahler-Einstein metrics
• Unnormalize conical Kähler-Ricci flow
• Ricci flow from spaces with isolated conical singularities
• The Yamabe flow on incomplete manifolds
• Convergence of the conical Ricci flow on S2 to a soliton
• A Survey on the Ricci flow on Singular Spaces
• Ricci de Turck flow on singular manifolds
• On the regularity problem of complex Monge-Ampere equations with conical singularities
• $C^{2,\alpha }$-estimate for conical Kähler-Ricci flow
• Stability of Ricci de Turck flow on Singular Spaces
• The conical Kähler-Ricci flow with weak initial data on Fano manifold
• Schauder estimates for equations with cone metrics, II
• $C^{2,\alpha }$-estimate for Monge-Ampere equations with Hölder-continuous right hand side
• Ricci flow on cone surfaces and a three-dimensional expanding soliton
• Ricci measure for some singular Riemannian metrics
• Smoothing conic Kähler metrics with uniformly upper bisectional curvature bound
• Bounded Ricci Curvature and Positive Scalar Curvature under Singular Ricci de Turck Flow
• The Kähler-Ricci flow on compact Kähler manifolds