Displaying similar documents to “The Geometry of Differential Harnack Estimates”

Global pinching theorems for minimal submanifolds in spheres

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 S n + p ( 1 ) . By using the Sobolev inequalities of P. Li to get L p estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and | | σ | | p the mean curvature and the L p norm of the square length of the second fundamental form of M. We show that there is a constant C such that if | | σ | | n / 2 < C , then M is a minimal submanifold in the sphere S n + p - 1 ( 1 + H ² ) with sectional...

Regularity of stable solutions of p -Laplace equations through geometric Sobolev type inequalities

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 Δ p u = g ( u ) in a smooth bounded domain Ω n . In particular, we obtain new L r and W 1 , r bounds for the extremal solution u when the domain is strictly convex. More precisely, we prove that u L ( Ω ) if n p + 2 and u L n p n - p - 2 ( Ω ) W 0 1 , p ( Ω ) if n > p + 2 .

Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

Mouhamed Moustapha Fall, Fethi Mahmoudi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Given a domain Ω of m + 1 and a k -dimensional non-degenerate minimal submanifold K of Ω with 1 k m - 1 , we prove the existence of a family of embedded constant mean curvature hypersurfaces in Ω which as their mean curvature tends to infinity concentrate along K and intersecting Ω perpendicularly along their boundaries.

Mean curvature properties for p -Laplace phase transitions

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 p -Laplacian type and a double well potential h 0 with suitable growth conditions. We prove that level sets of solutions of Δ p u = h 0 ' ( u ) 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.

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

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 z = - c t + φ ( x ) , φ : x N - 1 φ ( x ) , N 2 , solutions to the N -dimensional forced mean curvature motion V n = - c 0 + κ ( c c 0 ) with prescribed asymptotics. For any 1 -homogeneous function φ , viscosity solution to the eikonal equation | D φ | = ( c / c 0 ) 2 - 1 , we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by  φ . We also describe φ in terms of a probability measure on  § N - 2 .

Two-dimensional curvature functionals with superquadratic growth

Ernst Kuwert, Tobias Lamm, Yuxiang Li (2015)

Journal of the European Mathematical Society

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For two-dimensional, immersed closed surfaces f : Σ n , we study the curvature functionals p ( f ) and 𝒲 p ( f ) with integrands ( 1 + | A | 2 ) p / 2 and ( 1 + | H | 2 ) p / 2 , respectively. Here A is the second fundamental form, H is the mean curvature and we assume p > 2 . Our main result asserts that W 2 , p critical points are smooth in both cases. We also prove a compactness theorem for 𝒲 p -bounded sequences. In the case of p this is just Langer’s theorem [16], while for 𝒲 p we have to impose a bound for the Willmore energy strictly below 8 π as an additional...

Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

Nguyen Ngoc Khanh (2016)

Archivum Mathematicum

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In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds ( M , g ) for the following general heat equation u t = Δ V u + a u log u + b u where a is a constant and b is a differentiable function defined on M × [ 0 , ) . We suppose that the Bakry-Émery curvature and the N -dimensional Bakry-Émery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently. ...

On Schrödinger maps from T 1 to  S 2

Robert L. Jerrard, Didier Smets (2012)

Annales scientifiques de l'École Normale Supérieure

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We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from T 1 to  S 2 . This estimate yields some continuity properties of the flow map for the topology of  L 2 ( T 1 , S 2 ) , provided one takes its quotient by the continuous group action of  T 1 given by translations. We also prove that without taking this quotient, for any t &gt; 0 the flow map at time t is discontinuous as a map from 𝒞 ( T 1 , S 2 ) , equipped with the weak topology of  H 1 / 2 , to the space of distributions ( 𝒞 ( T 1 , 3 ) ) * . The argument relies...

A geometric problem and the Hopf Lemma. I

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 n -dimensional manifold without boundary, embedded in n + 1 , and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane X n + 1 = const in case M satisfies: for any two points ( X ' , X n + 1 ) , ( X ' , X ^ n + 1 ) on M , with X n + 1 > X ^ n + 1 , 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 n = 1 ....

Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Zdeněk Dušek (2015)

Czechoslovak Mathematical Journal

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Let ( M , g ) be a 4-dimensional Einstein Riemannian manifold. At each point p of M , the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor R at p . In this basis, up to standard symmetries and antisymmetries, just 5 components of the curvature tensor R are nonzero. For the space of constant curvature, the group O ( 4 ) acts as a transformation group between ST bases at T p M and for the so-called 2-stein curvature tensors, the group Sp ( 1 ) SO ( 4 ) acts as a transformation...

A strong maximum principle for the Paneitz operator and a non-local flow for the Q -curvature

Matthew J. Gursky, Andrea Malchiodi (2015)

Journal of the European Mathematical Society

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In this paper we consider Riemannian manifolds ( M n , g ) of dimension n 5 , with semi-positive Q -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 Q -curvature. Modifying the test function construction of Esposito-Robert,...

On Kakeya–Nikodym averages, L p -norms and lower bounds for nodal sets of eigenfunctions in higher dimensions

Matthew D. Blair, Christopher D. Sogge (2015)

Journal of the European Mathematical Society

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We extend a result of the second author [27, Theorem 1.1] to dimensions d 3 which relates the size of L p -norms of eigenfunctions for 2 < p < 2 ( d + 1 ) / d - 1 to the amount of L 2 -mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an " ϵ removal lemma" of Tao and Vargas [35]. We also use Hörmander’s [20] L 2 oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive...

A new characterization of the sphere in R 3

Thomas Hasanis (1980)

Annales Polonici Mathematici

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Let M be a closed connected surface in R 3 with positive Gaussian curvature K and let K I I be the curvature of its second fundamental form. It is shown that M is a sphere if K I I = c H K r , for some constants c and r, where H is the mean curvature of M.

A characterization of n-dimensional hypersurfaces in R n + 1 with commuting curvature operators

Yulian T. Tsankov (2005)

Banach Center Publications

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Let Mⁿ be a hypersurface in R n + 1 . We prove that two classical Jacobi curvature operators J x and J y 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 ( K x , y K z , u ) ( u ) = ( K z , u K x , y ) ( u ) , where K x , y ( u ) = R ( x , y , u ) , for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.

Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures

Eric A. Carlen, Dario Cordero-Erausquin, Elliott H. Lieb (2013)

Annales de l'I.H.P. Probabilités et statistiques

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An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the L 2 norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site...

Geometry and inequalities of geometric mean

Trung Hoa Dinh, Sima Ahsani, Tin-Yau Tam (2016)

Czechoslovak Mathematical Journal

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We study some geometric properties associated with the t -geometric means A t B : = A 1 / 2 ( A - 1 / 2 B A - 1 / 2 ) t A 1 / 2 of two n × n positive definite matrices A and B . Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding...

Local gradient estimates of p -harmonic functions, 1 / H -flow, and an entropy formula

Brett Kotschwar, Lei Ni (2009)

Annales scientifiques de l'École Normale Supérieure

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In the first part of this paper, we prove local interior and boundary gradient estimates for p -harmonic functions on general Riemannian manifolds. With these estimates, following the strategy in recent work of R. Moser, we prove an existence theorem for weak solutions to the level set formulation of the 1 / H (inverse mean curvature) flow for hypersurfaces in ambient manifolds satisfying a sharp volume growth assumption. In the second part of this paper, we consider two parabolic analogues...

Pointed k -surfaces

Graham Smith (2006)

Bulletin de la Société Mathématique de France

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Let S be a Riemann surface. Let 3 be the 3 -dimensional hyperbolic space and let 3 be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping ϕ : S 3 = ^ . If i : S 3 is a convex immersion, and if N is its exterior normal vector field, we define the Gauss lifting, ı ^ , of i by ı ^ = N . Let n : U 3 3 be the Gauss-Minkowski mapping. A solution to the Plateau problem ( S , ϕ ) is a convex immersion i of constant Gaussian curvature equal to k ( 0 , 1 ) such that the Gauss lifting ( S , ı ^ ) is complete and n ı ^ = ϕ . In this...