Results 1  10
of
118
Manifolds with positive curvature operator are space forms
 ANN. OF MATH
"... ... that a compact threemanifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact fourmanifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for ..."
Abstract

Cited by 115 (2 self)
 Add to MetaCart
... that a compact threemanifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact fourmanifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact fourmanifolds with 2positive curvature operators [Che]. Recall that a curvature operator is called 2positive, if the sum of its two smallest eigenvalues is positive. In arbitrary dimensions Huisken [Hu] described an explicit open cone in the space of curvature operators such that the normalized Ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature. Hamilton conjectured that in all dimensions compact Riemannian manifolds with positive curvature operators must be space forms. In this paper we confirm this conjecture. More generally, we show the following Theorem 1. On a compact manifold the normalized Ricci flow evolves a Riemannian metric with 2positive curvature operator to a limit metric with constant sectional curvature. The theorem is known in dimensions below five [H3], [H1], [Che]. Our proof works in dimensions above two: we only use Hamilton’s maximum principle and Klingenberg’s injectivity radius estimate for quarter pinched manifolds. Since in dimensions above two a quarter pinched orbifold is covered by a manifold (see Proposition 5.2), our proof carries over to orbifolds. This is no longer true in dimension two. In the manifold case it is known that the normalized Ricci flow converges to a metric of constant curvature for any initial metric [H3], [Cho]. However, there exist twodimensional orbifolds with positive sectional curvature which are not covered by a manifold. On such orbifolds the Ricci flow converges to a nontrivial Ricci soliton [CW]. Let us mention that a 2positive curvature operator has positive isotropic curvature. Micallef and Moore [MM] showed that a simply connected compact manifold with positive isotropic curvature is a homotopy sphere. However, their techniques do not allow to get restrictions for the fundamental groups or the differentiable structure of the underlying manifold.
Combinatorial Ricci flows on surfaces
 JOURNAL OF DIFFERENTIAL GEOMETRY
, 2003
"... ..."
(Show Context)
A fully nonlinear conformal flow on locally conformally flat manifolds
 JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
, 2001
"... We study a fully nonlinear flow for conformal metrics. The longtime existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the σkYamabe problem for locally conformal flat manifolds when k ̸ = n/2. ..."
Abstract

Cited by 64 (14 self)
 Add to MetaCart
(Show Context)
We study a fully nonlinear flow for conformal metrics. The longtime existence and the sequential convergence of flow are established for locally conformally flat manifolds. As an application, we solve the σkYamabe problem for locally conformal flat manifolds when k ̸ = n/2.
Discrete Surface Ricci Flow
 SUBMITTED TO IEEE TVCG
"... This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by userdefined Gaussian curvatures. Furthermore, the target metrics are conform ..."
Abstract

Cited by 40 (22 self)
 Add to MetaCart
This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by userdefined Gaussian curvatures. Furthermore, the target metrics are conformal (anglepreserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton’s method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.
The entropy formula for linear heat equation
 J. Geom. Anal
, 2004
"... ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the wellknown Li–Yau’s gradient estimate. As a byproduct we obtain the sharp ..."
Abstract

Cited by 38 (11 self)
 Add to MetaCart
(Show Context)
ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the wellknown Li–Yau’s gradient estimate. As a byproduct we obtain the sharp estimates on ‘Nash’s entropy ’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li–Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to R n. In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman’s entropy in the case of Riemann surfaces. 1. The relation with Li–Yau’s gradient estimates In this section we provide another derivation of Theorem 1.1 of [8] and discuss its relation with Li–Yau’s gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash’s ‘entropy quantity ’ − � M H log Hdvin the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of [9]. Let u(x, t) be a positive solution to � ∂ ∂t − � � u(x, t) = 0 with �
The logrithmic Sobolev inequality along the Ricci flow, arXiv:0707.2424v4
"... 2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow ..."
Abstract

Cited by 33 (2 self)
 Add to MetaCart
2. The Sobolev inequality 3. The logarithmic Sobolev inequality on a Riemannian manifold 4. The logarithmic Sobolev inequality along the Ricci flow 5. The Sobolev inequality along the Ricci flow
The KählerRicci flow with positive bisectional curvature
 Invent. Math
"... We show that the KählerRicci flow on a manifold with positive first Chern class converges to a KählerEinstein metric assuming positive bisectional curvature and certain stability conditions. 1 ..."
Abstract

Cited by 21 (3 self)
 Add to MetaCart
(Show Context)
We show that the KählerRicci flow on a manifold with positive first Chern class converges to a KählerEinstein metric assuming positive bisectional curvature and certain stability conditions. 1
MODULAR CURVATURE FOR NONCOMMUTATIVE TWOTORI
, 1110
"... Abstract. Starting from the description of the conformal geometry of noncommutative 2tori in the framework of modular spectral triples, we explicitly compute the local curvature functionals determined by the value at zero of the zeta functions affiliated with these spectral triples. We give a close ..."
Abstract

Cited by 21 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Starting from the description of the conformal geometry of noncommutative 2tori in the framework of modular spectral triples, we explicitly compute the local curvature functionals determined by the value at zero of the zeta functions affiliated with these spectral triples. We give a closed formula for the RaySinger analytic torsion in terms of the Dirichlet quadratic form and the generating function for Bernoulli numbers applied to the modular operator. The gradient of the RaySinger analytic torsion is then expressed in terms of these functionals, and yields the analogue of scalar curvature. Computing this gradient in two ways elucidates the meaning of the complicated two variable functions occurring in the formula for the scalar curvature. Moreover, the corresponding evolution equation for the metric produces the appropriate analogue of Ricci curvature. We prove the analogue of the classical result which asserts that in every conformal class the maximum value of the determinant of the Laplacian on metrics of a fixed area is attained only at the constant curvature metric.
Ricci flow and the determinant of the Laplacian on noncompact surfaces
 Comm. Par. Diff. Eq
, 2013
"... ar ..."
(Show Context)
Stability of Euclidean space under Ricci flow
 Comm. Anal. Geom
"... Abstract. We study the Ricci flow for initial metrics which are C 0 small perturbations of the Euclidean metric on R n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We study the Ricci flow for initial metrics which are C 0 small perturbations of the Euclidean metric on R n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In proving this stability result, we introduce a monotone integral quantity which measures the deviation of the evolving metric from the Euclidean metric. We also investigate the convergence of the diffeomorphisms relating Ricci harmonic map heat flow to Ricci flow. 1.