$Q$-curvature and spectral invariants

Branson, Thomas

Displaying similar documents to “ $Q$ -curvature and spectral invariants”

On the spectral radius in $L_{1} (G)$

E. Porada (1971)

Colloquium Mathematicae

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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 $K_{I} I = c \sqrt H K^{r}$ , for some constants c and r, where H is the mean curvature of M.

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.

A spectral gap theorem in SU $(d)$

Jean Bourgain, Alex Gamburd (2012)

Journal of the European Mathematical Society

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We establish the spectral gap property for dense subgroups of SU $(d)$ $(d \geq 2)$ , generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU $(2)$ .

Spectral synthesis and operator synthesis

K. Parthasarathy, R. Prakash (2006)

Studia Mathematica

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Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty} (G)$ -submodule X̂ of ℬ(L²(G)) (where $V^{\infty} (G)$ is the weak-* Haagerup tensor product $L^{\infty} (G) \otimes_{w * h} L^{\infty} (G)$ ), define the concept of X̂-operator synthesis and prove that a...

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 $f : Σ \to ℝ^{n}$ , we study the curvature functionals $ℰ^{p} (f)$ 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 $ℰ^{p}$ 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...

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 $Sp (1) \subset SO (4)$ acts as a transformation...

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 $ℝ^{n + 1}$ , 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 $(X^{'}, X_{n + 1})$ , $(X^{'}, {\hat{X}}_{n + 1})$ on $M$ , with $X_{n + 1} > {\hat{X}}_{n + 1}$ , 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 $n = 1$ ....

The resolution of the bounded $L^{2}$ curvature conjecture in general relativity

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 $L^{2}$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^{2}$ -norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.

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...

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 $ℝ^{m + 1}$ and a $k$ -dimensional non-degenerate minimal submanifold $K$ of $\partial Ω$ with $1 \leq k \leq m - 1$ , 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 $\partial Ω$ perpendicularly along their boundaries.

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

Spectral projections for the twisted Laplacian

Herbert Koch, Fulvio Ricci (2007)

Studia Mathematica

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Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L = - 1 / 2 \sum_{j = 1}^{n} [(\partial_{x_{j}} + i y_{j}) ² + (\partial_{y_{j}} - i x_{j}) ²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{λ}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $| | P_{λ} {u | |}_{L^{p} (ℝ^{d})} ≲ λ^{ϱ (p)} {| | u | |}_{L ² (ℝ^{d})}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p...

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 $(K_{x, y} \circ K_{z, u}) (u) = (K_{z, u} \circ K_{x, y}) (u)$ , where $K_{x, y} (u) = R (x, y, u)$ , for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.

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 $Ω \subset ℝ^{n}$ . 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 $u^{☆} \in L^{\infty} (Ω)$ if $n \leq p + 2$ and $u^{☆} \in L^{\frac{n p}{n - p - 2}} (Ω) \cap W_{0}^{1, p} (Ω)$ if $n > p + 2$ .

Spectral radius of operators associated with dynamical systems in the spaces C(X)

Krzysztof Zajkowski (2005)

Banach Center Publications

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We consider operators acting in the space C(X) (X is a compact topological space) of the form $A u (x) = (\sum_{k = 1}^{N} e^{φ_{k}} T_{α_{k}}) u (x) = \sum_{k = 1}^{N} e^{φ_{k} (x)} u (α_{k} (x))$ , u ∈ C(X), where $φ_{k} \in C (X)$ and $α_{k} : X \to X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $φ = {(φ_{k})}_{k = 1}^{N}$ . We prove that $l n (r (A)) = λ (φ) = m a x_{ν \in M e s} \sum_{k = 1}^{N} \int_{X} φ_{k} d ν_{k} - λ * (ν)$ , where Mes is the set of all probability vectors of measures $ν = {(ν_{k})}_{k = 1}^{N}$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on...

Transferring $L^{p}$ eigenfunction bounds from $S^{2 n + 1}$ to hⁿ

Valentina Casarino, Paolo Ciatti (2009)

Studia Mathematica

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By using the notion of contraction of Lie groups, we transfer $L^{p} - L ²$ estimates for joint spectral projectors from the unit complex sphere $S^{2 n + 1}$ in $ℂ^{n + 1}$ to the reduced Heisenberg group hⁿ. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on hⁿ. As a consequence, we prove, in the spirit of Sogge’s work, a discrete restriction theorem for the sub-Laplacian L on hⁿ.

On some properties of a Riemannian space $V_{4}$ with constant scalar curvature

W. Slósarska, Z. Żekanowski (1972)

Colloquium Mathematicae

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The Salvetti complex and the little cubes

Dai Tamaki (2012)

Journal of the European Mathematical Society

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For a real central arrangement $𝒜$ , Salvetti introduced a construction of a finite complex Sal $(𝒜)$ which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement $𝒜_{k - 1}$ , the Salvetti complex Sal $(𝒜_{k - 1})$ serves as a good combinatorial model for the homotopy type of the configuration space $F (ℂ, k)$ of $k$ points in $C$ , which is homotopy equivalent to the space $C_{2} (k)$ of k little $2$ -cubes. Motivated by the importance of little cubes in homotopy theory, especially in...

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 $d \geq 3$ 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...

Displaying similar documents to “ Q -curvature and spectral invariants”

Displaying similar documents to “ $Q$ -curvature and spectral invariants”