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Search: MSC category 53C21 ( Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] )

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1. CJM 2016 (vol 68 pp. 655)

Klartag, Bo'az; Kozma, Gady; Ralli, Peter; Tetali, Prasad
 Discrete Curvature and Abelian Groups We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of curvature'' in discrete spaces. An appealing feature of this discrete version of the so-called $\Gamma_2$-calculus (of Bakry-Ãmery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest -- particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (Ã  la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs -- a result of independent interest. Keywords:Ricci curvature, graph theory, abelian groupsCategories:53C21, 57M15

2. CJM 2012 (vol 65 pp. 266)

Bérard, Vincent
 Les applications conforme-harmoniques Sur une surface de Riemann, l'Ã©nergie d'une application Ã  valeurs dans une variÃ©tÃ© riemannienne est une fonctionnelle invariante conforme, ses points critiques sont les applications harmoniques. Nous proposons ici un analogue en dimension supÃ©rieure, en construisant une fonctionnelle invariante conforme pour les applications entre deux variÃ©tÃ©s riemanniennes, dont la variÃ©tÃ© de dÃ©part est de dimension $n$ paire. Ses points critiques satisfont une EDP elliptique d'ordre $n$ non-linÃ©aire qui est covariante conforme par rapport Ã  la variÃ©tÃ© de dÃ©part, on les appelle les applications conforme-harmoniques. Dans le cas des fonctions, on retrouve l'opÃ©rateur GJMS, dont le terme principal est une puissance $n/2$ du laplacien. Quand $n$ est impaire, les mÃªmes idÃ©es permettent de montrer que le terme constant dans le dÃ©veloppement asymptotique de l'Ã©nergie d'une application asymptotiquement harmonique sur une variÃ©tÃ© AHE est indÃ©pendant du choix du reprÃ©sentant de l'infini conforme. Categories:53C21, 53C43, 53A30

3. CJM 2012 (vol 65 pp. 757)

Delanoë, Philippe; Rouvière, François
 Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved The squared distance curvature is a kind of two-point curvature the sign of which turned out crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, an indirect one via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space. Keywords:symmetric spaces, rank one, positive curvature, almost-positive $c$-curvatureCategories:53C35, 53C21, 53C26, 49N60

4. CJM 2011 (vol 64 pp. 778)

Calvaruso, Giovanni; Fino, Anna
 Ricci Solitons and Geometry of Four-dimensional Non-reductive Homogeneous Spaces We study the geometry of non-reductive $4$-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable. Keywords:non-reductive homogeneous spaces, pseudo-Riemannian metrics, Ricci solitons, Einstein-like metricsCategories:53C21, 53C50, 53C25

5. CJM 2010 (vol 62 pp. 1264)

Chen, Jingyi; Fraser, Ailana
 Holomorphic variations of minimal disks with boundary on a Lagrangian surface Let $L$ be an oriented Lagrangian submanifold in an $n$-dimensional KÃ¤hler manifold~$M$. Let $u \colon D \to M$ be a minimal immersion from a disk $D$ with $u(\partial D) \subset L$ such that $u(D)$ meets $L$ orthogonally along $u(\partial D)$. Then the real dimension of the space of admissible holomorphic variations is at least $n+\mu(E,F)$, where $\mu(E,F)$ is a boundary Maslov index; the minimal disk is holomorphic if there exist $n$ admissible holomorphic variations that are linearly independent over $\mathbb{R}$ at some point $p \in \partial D$; if $M = \mathbb{C}P^n$ and $u$ intersects $L$ positively, then $u$ is holomorphic if it is stable, and its Morse index is at least $n+\mu(E,F)$ if $u$ is unstable. Categories:58E12, 53C21, 53C26

6. CJM 2005 (vol 57 pp. 708)

Finster, Felix; Kraus, Margarita
 Curvature Estimates in Asymptotically Flat Lorentzian Manifolds We consider an asymptotically flat Lorentzian manifold of dimension $(1,3)$. An inequality is derived which bounds the Riemannian curvature tensor in terms of the ADM energy in the general case with second fundamental form. The inequality quantifies in which sense the Lorentzian manifold becomes flat in the limit when the ADM energy tends to zero. Categories:53C21, 53C27, 83C57
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