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1
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The Cotton Tensor and the Ricci Flow

100%
Mantegazza C., Mongodi S., Rimoldi M.
Geometric Flows
|
2017
|
tom 2
|
nr 1
EN
We compute the evolution equation of the Cotton and the Bach tensor under the Ricci flow of a Riemannian manifold, with particular attention to the three dimensional case, and we discuss some applications.
2
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On Asymmetric Distances

76%
Mennucci A. C. G.
Analysis and Geometry in Metric Spaces
|
2013
|
tom 1
200-231
EN
In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.
3
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Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

76%
Mendel M., Naor A.
Analysis and Geometry in Metric Spaces
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2013
|
tom 1
163-199
EN
The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.
4
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Resolvent Flows for Convex Functionals and p-Harmonic Maps

76%
Kuwae K.
Analysis and Geometry in Metric Spaces
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2015
|
tom 3
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nr 1
EN
We prove the unique existence of the (non-linear) resolvent associated to a coercive proper lower semicontinuous function satisfying a weak notion of p-uniform λ-convexity on a complete metric space, and establish the existence of the minimizer of such functions as the large time limit of the resolvents, which generalizing pioneering work by Jost for convex functionals on complete CAT(0)-spaces. The results can be applied to Lp-Wasserstein space over complete p-uniformly convex spaces. As an application, we solve an initial boundary value problem for p-harmonic maps into CAT(0)-spaces in terms of Cheeger type p-Sobolev spaces.
5
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Gradient estimates for inverse curvature flows in hyperbolic space

76%
Scheuer J.
Geometric Flows
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2015
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tom 1
|
nr 1
EN
We prove gradient estimates for hypersurfaces in the hyperbolic space Hn+1, expanding by negative powers of a certain class of homogeneous curvature functions F. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers p > 1 of F-1 and smooth convergence of the properly rescaled hypersurfaces. In particular, the full convergence result holds for the inverse Gauss curvature flow of surfaces without any further pinching condition besides convexity of the initial hypersurface.
6
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Topological tools for the prescribed scalar curvature problem on S n

64%
Abuzaid D., Ben Mahmoud R., Chtioui H., Rigane A.
Open Mathematics
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2014
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tom 12
|
nr 12
1829-1839
EN
In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].
7
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Sobolev-Kantorovich Inequalities

52%
Ledoux M.
Analysis and Geometry in Metric Spaces
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2015
|
tom 3
|
nr 1
EN
In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.
8
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Dirac and Plateau billiards in domains with corners

52%
Gromov M.
Open Mathematics
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2014
|
tom 12
|
nr 8
1109-1156
EN
Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C 2-smooth Riemannian metrics g on a smooth manifold X, such that scalg(x) ≥ κ(x), is closed under C 0-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.
9
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Affine compact almost-homogeneous manifolds of cohomogeneity one

52%
Guan D.
Open Mathematics
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2009
|
tom 7
|
nr 1
84-123
EN
This paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.
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