Manifolds with quadratic curvature decay and slow volume growth
Manifolds with singularities accepting a metric of positive scalar curvature.
Mean curvature comparison for tubular hypersurfaces in symmetric spaces.
Means in complete manifolds: uniqueness and approximation
Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure μ ( x ) = 1 N ∑ k = 1 N δ x k has a uniquep–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is...
Meilleures constantes dans le théorème d'inclusion de Sobolev
Meilleures constantes et inégalités de Sobolev optimales sur les variétés riemanniennes compactes
Metric Ricci Curvature and Flow for PL Manifolds
We summarize here the main ideas and results of our papers [28], [14], as presented at the 2013 CIRM Meeting on Discrete curvature and we augment these by bringing up an application of one of our main results, namely to solving a problem regarding cube complexes.
Metric tensor estimates, geometric convergence, and inverse boundary problems.
Metrics of constant curvature on a Riemann surface with two corners on the boundary
Metrics with Harmonic Spinors.
Microlocal Approach to Tensor Tomography and Boundary and Lens Rigidity
2000 Mathematics Subject Classification: 53C24, 53C65, 53C21.This is a survey of the recent results by the author and Gunther Uhlmann on the boundary rigidity problem and on the associated tensor tomography problem.Author partly supported by NSF Grant DMS-0400869.
Minimal submanifolds in with a g.c.K. structure
In this paper we obtain all invariant, anti-invariant and submanifolds in endowed with a globally conformal Kähler structure which are minimal and tangent or normal to the Lee vector field of the g.c.K. structure.
Minimal surfaces, the Dirac operator and the Penrose inequality
Minoration du spectre des variétés hyperboliques de dimension 3
Soit une variété hyperbolique compacte de dimension 3, de diamètre et de volume . Si on note la -ième valeur propre du laplacien de Hodge-de Rham agissant sur les 1-formes coexactes de , on montre que et , où est une constante ne dépendant que de , et est le nombre de composantes connexes de la partie mince de . En outre, on montre que pour toute 3-variété hyperbolique de volume fini avec cusps, il existe une suite de remplissages compacts de , de diamètre telle que et .
Minorations sur le des variétés riemanniennes
Moduli space, heights and isospectral sets of metrics
Monge-Ampère equations and surfaces with negative Gaussian curvature
In [24], we studied the singularities of solutions of Monge-Ampère equations of hyperbolic type. Then we saw that the singularities of solutions do not coincide with the singularities of solution surfaces. In this note we first study the singularities of solution surfaces. Next, as the applications, we consider the singularities of surfaces with negative Gaussian curvature. Our problems are as follows: 1) What kinds of singularities may appear?, and 2) How can we extend the surfaces beyond the singularities?...
Monogenic functions in conformal geometry.
Monopole metrics and the orbifold Yamabe problem
We consider the self-dual conformal classes on discovered by LeBrun. These depend upon a choice of points in hyperbolic -space, called monopole points. We investigate the limiting behavior of various constant scalar curvature metrics in these conformal classes as the points approach each other, or as the points tend to the boundary of hyperbolic space. There is a close connection to the orbifold Yamabe problem, which we show is not always solvable (in contrast to the case of compact manifolds)....
Monotone point-to-set vector fields.