A Study on -recurrence -curvature tensor in -contact metric manifolds
Gurupadavva Ingalahalli, C.S. Bagewadi (2018)
Communications in Mathematics
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In this paper we study -recurrence -curvature tensor in-contact metric manifolds.
Gurupadavva Ingalahalli, C.S. Bagewadi (2018)
Communications in Mathematics
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In this paper we study -recurrence -curvature tensor in-contact metric manifolds.
Thomas Hasanis (1980)
Annales Polonici Mathematici
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Let M be a closed connected surface in with positive Gaussian curvature K and let be the curvature of its second fundamental form. It is shown that M is a sphere if , for some constants c and r, where H is the mean curvature of M.
Abderrahim Zagane, Mustapha Djaa (2018)
Communications in Mathematics
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In this paper, we introduce the Mus-Sasaki metric on the tangent bundle as a new natural metric non-rigid on . First we investigate the geometry of the Mus-Sasakian metrics and we characterize the sectional curvature and the scalar curvature.
Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel (2014-2015)
Séminaire Laurent Schwartz — EDP et applications
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This paper reports on the recent proof of the bounded curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the -norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.
Kairen Cai (2003)
Colloquium Mathematicae
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Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere . By using the Sobolev inequalities of P. Li to get estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and the mean curvature and the norm of the square length of the second fundamental form of M. We show that there is a constant C such that if , then M is a minimal submanifold in the sphere with sectional...
Berardino Sciunzi, Enrico Valdinoci (2005)
Journal of the European Mathematical Society
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This paper deals with phase transitions corresponding to an energy which is the sum of a kinetic part of -Laplacian type and a double well potential with suitable growth conditions. We prove that level sets of solutions of possessing a certain decay property satisfy a mean curvature equation in a suitable weak viscosity sense. From this, we show that, if the above level sets approach uniformly a hypersurface, the latter has zero mean curvature.
Zdeněk Dušek (2015)
Czechoslovak Mathematical Journal
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Let be a 4-dimensional Einstein Riemannian manifold. At each point of , the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor at . In this basis, up to standard symmetries and antisymmetries, just components of the curvature tensor are nonzero. For the space of constant curvature, the group acts as a transformation group between ST bases at and for the so-called 2-stein curvature tensors, the group acts as a transformation...
Matthew J. Gursky, Andrea Malchiodi (2015)
Journal of the European Mathematical Society
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In this paper we consider Riemannian manifolds of dimension , with semi-positive -curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive -curvature. Modifying the test function construction of Esposito-Robert,...
Ernst Kuwert, Tobias Lamm, Yuxiang Li (2015)
Journal of the European Mathematical Society
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For two-dimensional, immersed closed surfaces , we study the curvature functionals and with integrands and , respectively. Here is the second fundamental form, is the mean curvature and we assume . Our main result asserts that critical points are smooth in both cases. We also prove a compactness theorem for -bounded sequences. In the case of this is just Langer’s theorem [16], while for we have to impose a bound for the Willmore energy strictly below as an additional...
Oihane F. Blanco, Miguel Sánchez, José M. Senovilla (2013)
Journal of the European Mathematical Society
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, that is to say, Lorentzian manifolds with vanishing second derivative of the curvature tensor , are characterized by several geometric properties, and explicitly presented. Locally, they are a product where each factor is uniquely determined as follows: is a Riemannian symmetric space and is either a constant-curvature Lorentzian space or a definite type of plane wave generalizing the Cahen–Wallach family. In the proper case (i.e., at some point), the curvature...
W. Slósarska, Z. Żekanowski (1972)
Colloquium Mathematicae
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Neil Seshadri (2009)
Bulletin de la Société Mathématique de France
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To any smooth compact manifold endowed with a contact structure and partially integrable almost CR structure , we prove the existence and uniqueness, modulo high-order error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric on . We consider the asymptotic expansion, in powers of a special defining function, of the volume of with respect to and prove that the log term coefficient is independent of (and any choice...
Yan Yan Li, Louis Nirenberg (2006)
Journal of the European Mathematical Society
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A classical result of A. D. Alexandrov states that a connected compact smooth -dimensional manifold without boundary, embedded in , and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of in a hyperplane in case satisfies: for any two points , on , with , the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for ....
Régis Monneau, Jean-Michel Roquejoffre, Violaine Roussier-Michon (2013)
Annales scientifiques de l'École Normale Supérieure
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We construct travelling wave graphs of the form , , , solutions to the -dimensional forced mean curvature motion () with prescribed asymptotics. For any -homogeneous function , viscosity solution to the eikonal equation , we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by . We also describe in terms of a probability measure on .
Daniele Castorina, Manel Sanchón (2015)
Journal of the European Mathematical Society
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We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of in a smooth bounded domain . In particular, we obtain new and bounds for the extremal solution when the domain is strictly convex. More precisely, we prove that if and if .
Mouhamed Moustapha Fall, Fethi Mahmoudi (2008)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Given a domain of and a -dimensional non-degenerate minimal submanifold of with , we prove the existence of a family of embedded constant mean curvature hypersurfaces in which as their mean curvature tends to infinity concentrate along and intersecting perpendicularly along their boundaries.
Yulian T. Tsankov (2005)
Banach Center Publications
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Let Mⁿ be a hypersurface in . We prove that two classical Jacobi curvature operators and commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation , where , for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.