Results 1  10
of
20
Continuity, curvature, and the general covariance of optimal transportation
, 2008
"... Let M and ¯ M be ndimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ; the ..."
Abstract

Cited by 75 (20 self)
 Add to MetaCart
Let M and ¯ M be ndimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via nonnegativity of the sectional curvature of certain nullplanes in a novel but natural pseudoRiemannian geometry which the cost c induces on the product space M × ¯ M. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry. Using the pseudoRiemannian structure, we extend Ma, Trudinger and Wang’s conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principal characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and ¯ M of Rn. This maximum principle plays a key role in Loeper’s Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered.
Local gradient estimates of solutions to some conformally invariant fully nonlinear equations
, 2006
"... ..."
(Show Context)
Viscosity solutions to second order partial differential equations on riemannian manifolds, ArXiv:math.AP/0612742v2
, 2007
"... Abstract. We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F(x, u(x), du(x), d 2 u(x)) = 0 defined on a finitedimensional Riemannian manifold M. Finest results (with hypothesis that require ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F(x, u(x), du(x), d 2 u(x)) = 0 defined on a finitedimensional Riemannian manifold M. Finest results (with hypothesis that require the function F to be degenerate elliptic, that is nonincreasing in the second order derivative variable) are obtained under the assumption that M has nonnegative sectional curvature, while, if one additionally requires F to depend on d 2 u in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature. 1.
On the σ2scalar curvature
 J. Differential Geom
, 2010
"... Abstract. In this paper, we establish an analytic foundation for a fully nonlinear equation σ2 σ1 = f on manifolds with positive scalar curvature and apply it to give a (rough) classification of such manifolds. A crucial point is a simple observation that this equation is a degenerate elliptic equa ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper, we establish an analytic foundation for a fully nonlinear equation σ2 σ1 = f on manifolds with positive scalar curvature and apply it to give a (rough) classification of such manifolds. A crucial point is a simple observation that this equation is a degenerate elliptic equation without any condition on the sign of f and it is elliptic not only for f> 0 but also for f < 0. By defining a Yamabe constant Y2,1 with respect to this equation, we show that a positive scalar curvature manifold admits a conformal metric with positive scalar curvature and positive σ2scalar curvature if and only if Y2,1> 0. We give a complete solution for the corresponding Yamabe problem. Namely, let g0 be a positive scalar curvature metric, then in its conformal class there is a conformal metric with σ2(g) = κσ1(g), for some constant κ. Using these analytic results, we give a rough classification of the space of manifolds with positive scalar curvature metrics. 1.
Rigidity and stability of Einstein metrics for quadratic curvature functionals
, 1105
"... ar ..."
(Show Context)
On (2k)Minimal Submanifolds
, 706
"... Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total (2k)th GaussBonnet curvature function, called ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature is zero. It is classical that minimal submanifolds are the critical points of the volume function. In this paper, we examine the critical points of the total (2k)th GaussBonnet curvature function, called (2k)minimal submanifolds. We prove that they are characterized by the vanishing of a higher mean curvature, namely the (2k + 1)GaussBonnet curvature. Furthermore, we show that several properties of usual minimal submanifolds can be naturally generalized to (2k)minimal submanifolds.
Fourth order curvature flows and geometric applications, preprint (2010) ArXiv mathDG:1012.0342
"... We study a class of fourth order curvature flows on a compact Riemannian manifold, which includes the gradient flows of a number of quadratic geometric functionals, as for instance the L2 norm of the curvature. Such flows can develop a special kind of singularities, that could not appear in the Ricc ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
We study a class of fourth order curvature flows on a compact Riemannian manifold, which includes the gradient flows of a number of quadratic geometric functionals, as for instance the L2 norm of the curvature. Such flows can develop a special kind of singularities, that could not appear in the Ricci flow, namely singularities where the manifold collapses with bounded curvature. We show that this phenomenon cannot occur if we assume a uniform positive lower bound on the Yamabe invariant. In particular, for a number of gradient flows in dimension four, such a lower bound exists if we assume a bound on the initial energy. This implies that these flows can only develop singularities where the curvature blows up, and that blowingup sequences converge (up to a subsequence) to a “singularity model”, namely a complete Bachflat, scalarflat manifold. We prove a rigidity result for those model manifolds and show that if the initial energy is smaller than an explicit bound, then no singularity can occur. Under those assumptions, the flow exists for all time, and converges up to a subsequence to the sphere or the real projective space. This gives an alternative proof, under a slightly stronger assumption, of a result from Chang, Gursky and Yang asserting that integral pinched 4manifolds with positive Yamabe constant are space forms. 1
Complete conformal metrics of negative Ricci curvature on compact manifolds with boundary
 Int. Math. Res. Not. IMRN
"... We study the problem of finding complete conformal metrics determined by a symmetric function of Ricci tensor in a negative convex cone on compact manifolds. A consequence of our main results is that any smooth bounded domain in Euclidean space of dimension greater or equal to 3 admits a complete co ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
We study the problem of finding complete conformal metrics determined by a symmetric function of Ricci tensor in a negative convex cone on compact manifolds. A consequence of our main results is that any smooth bounded domain in Euclidean space of dimension greater or equal to 3 admits a complete conformally flatmetric of negative Ricci curvature with det(−Ric) = 1. 1