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Search: MSC category 58J ( Partial differential equations on manifolds; differential operators [See also 32Wxx, 35-XX, 53Cxx] )

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1. CJM 2014 (vol 67 pp. 107)

Chang, Jui-En; Xiao, Ling
 The Weyl Problem With Nonnegative Gauss Curvature In Hyperbolic Space In this paper, we discuss the isometric embedding problem in hyperbolic space with nonnegative extrinsic curvature. We prove a priori bounds for the trace of the second fundamental form $H$ and extend the result to $n$-dimensions. We also obtain an estimate for the gradient of the smaller principal curvature in 2 dimensions. Categories:53A99, 5J15, 58J05

2. CJM 2011 (vol 63 pp. 721)

Autin, Aymeric
 Isoresonant Complex-valued Potentials and Symmetries Let $X$ be a connected Riemannian manifold such that the resolvent of the free Laplacian $(\Delta-z)^{-1}$, $z\in\mathbb{C} \setminus \mathbb{R}^+$, has a meromorphic continuation through $\mathbb{R}^+$. The poles of this continuation are called resonances. When $X$ has some symmetries, we construct complex-valued potentials, $V$, such that the resolvent of $\Delta+V$, which has also a meromorphic continuation, has the same resonances with multiplicities as the free Laplacian. Categories:31C12, 58J50

3. CJM 2010 (vol 63 pp. 55)

Chau, Albert; Tam, Luen-Fai; Yu, Chengjie
 Pseudolocality for the Ricci Flow and Applications Perelman established a differential Li--Yau--Hamilton (LYH) type inequality for fundamental solutions of the conjugate heat equation corresponding to the Ricci flow on compact manifolds. As an application of the LYH inequality, Perelman proved a pseudolocality result for the Ricci flow on compact manifolds. In this article we provide the details for the proofs of these results in the case of a complete noncompact Riemannian manifold. Using these results we prove that under certain conditions, a finite time singularity of the Ricci flow must form within a compact set. The conditions are satisfied by asymptotically flat manifolds. We also prove a long time existence result for the K\"ahler--Ricci flow on complete nonnegatively curved K\"ahler manifolds. Categories:53C44, 58J37, 35B35

4. CJM 2009 (vol 61 pp. 548)

Girouard, Alexandre
 Fundamental Tone, Concentration of Density, and Conformal Degeneration on Surfaces We study the effect of two types of degeneration of a Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point within a conformal class. The second is degeneration of the conformal class to the boundary of the moduli space on the torus and on the Klein bottle. In the latter, we follow the outline proposed by N. Nadirashvili in 1996. Categories:35P, 58J

5. CJM 2008 (vol 60 pp. 1336)

Olver, Peter J.; Pohjanpelto, Juha
 Moving Frames for Lie Pseudo--Groups We propose a new, constructive theory of moving frames for Lie pseudo-group actions on submanifolds. The moving frame provides an effective means for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving equivalence and symmetry problems arising in a broad range of applications. Categories:58A15, 58A20, 58H05, 58J70

6. CJM 2008 (vol 60 pp. 572)

Hitrik, Michael; Sj{östrand, Johannes
 Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength $\epsilon$ of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$ (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point value of the flow average of the leading perturbation. Keywords:non-selfadjoint, eigenvalue, periodic flow, branching singularityCategories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40

7. CJM 2008 (vol 60 pp. 241)

Alexandrova, Ivana
 Semi-Classical Wavefront Set and Fourier Integral Operators Here we define and prove some properties of the semi-classical wavefront set. We also define and study semi-classical Fourier integral operators and prove a generalization of Egorov's theorem to manifolds of different dimensions. Keywords:wavefront set, Fourier integral operators, Egorov theorem, semi-classical analysisCategories:35S30, 35A27, 58J40, 81Q20

8. CJM 2008 (vol 60 pp. 457)

Teplyaev, Alexander
 Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure We define sets with finitely ramified cell structure, which are generalizations of post-crit8cally finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami's resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates. Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metricCategories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18

9. CJM 2007 (vol 59 pp. 943)

Finster, Felix; Kraus, Margarita
 A Weighted $L^2$-Estimate of the Witten Spinor in Asymptotically Schwarzschild Manifolds We derive a weighted $L^2$-estimate of the Witten spinor in a complete Riemannian spin manifold~$(M^n, g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of~$M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of $M$. Categories:83C60, 35Q75, 35J45, 58J05

10. CJM 2007 (vol 59 pp. 742)

Gil, Juan B.; Krainer, Thomas; Mendoza, Gerardo A.
 Geometry and Spectra of Closed Extensions of Elliptic Cone Operators We study the geometry of the set of closed extensions of index $0$ of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and we give a necessary and sufficient condition for the existence of rays of minimal growth for such operators. Keywords:resolvents, manifolds with conical singularities, spectral theor, boundary value problems, GrassmanniansCategories:58J50, 35J70, 14M15

11. CJM 2006 (vol 58 pp. 381)

Jakobson, Dmitry; Nadirashvili, Nikolai; Polterovich, Iosif
 Extremal Metric for the First Eigenvalue on a Klein Bottle The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus. We construct an extremal metric on a Klein bottle. It is a metric of revolution, admitting a minimal isometric embedding into a sphere ${\mathbb S}^4$ by the first eigenfunctions. Also, this Klein bottle is a bipolar surface for Lawson's $\tau_{3,1}$-torus. We conjecture that an extremal metric for the first eigenvalue on a Klein bottle is unique, and hence it provides a sharp upper bound for $\lambda_1$ on a Klein bottle of a given area. We present numerical evidence and prove the first results towards this conjecture. Keywords:Laplacian, eigenvalue, Klein bottleCategories:58J50, 53C42

12. CJM 2005 (vol 57 pp. 771)

Schrohe, E.; Seiler, J.
 The Resolvent of Closed Extensions of Cone Differential Operators We study closed extensions $\underline A$ of an elliptic differential operator $A$ on a manifold with conical singularities, acting as an unbounded operator on a weighted $L_p$-space. Under suitable conditions we show that the resolvent $(\lambda-\underline A)^{-1}$ exists in a sector of the complex plane and decays like $1/|\lambda|$ as $|\lambda|\to\infty$. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of $\underline A$. As an application we treat the Laplace--Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem $\dot{u}-\Delta u=f$, $u(0)=0$. Keywords:Manifolds with conical singularities, resolvent, maximal regularityCategories:35J70, 47A10, 58J40

13. CJM 2005 (vol 57 pp. 506)

Gross, Leonard; Grothaus, Martin
 Reverse Hypercontractivity for Subharmonic Functions Contractivity and hypercontractivity properties of semigroups are now well understood when the generator, $A$, is a Dirichlet form operator. It has been shown that in some holomorphic function spaces the semigroup operators, $e^{-tA}$, can be bounded {\it below} from $L^p$ to $L^q$ when $p,q$ and $t$ are suitably related. We will show that such lower boundedness occurs also in spaces of subharmonic functions. Keywords:Reverse hypercontractivity, subharmonicCategories:58J35, 47D03, 47D07, 32Q99, 60J35

14. CJM 2005 (vol 57 pp. 251)

Cocos, M.
 Some New Results on $L^2$ Cohomology of Negatively Curved Riemannian Manifolds The present paper is concerned with the study of the $L^2$ cohomology spaces of negatively curved manifolds. The first half presents a finiteness and vanishing result obtained under some curvature assumptions, while the second half identifies a class of metrics having non-trivial $L^2$ cohomology for degree equal to the half dimension of the space. For the second part we rely on the existence and regularity properties of the solution for the heat equation for forms. Category:58J50

15. CJM 2005 (vol 57 pp. 225)

Booss-Bavnbek, Bernhelm; Lesch, Matthias; Phillips, John
 Unbounded Fredholm Operators and Spectral Flow We study the gap (= projection norm'' = graph distance'') topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary. Categories:58J30, 47A53, 19K56, 58J32

16. CJM 2005 (vol 57 pp. 99)

Fegan, H. D.; Steer, B.
 Second Order Operators on a Compact Lie Group We describe the structure of the space of second order elliptic differential operators on a homogenous bundle over a compact Lie group. Subject to a technical condition, these operators are homotopic to the Laplacian. The technical condition is further investigated, with examples given where it holds and others where it does not. Since many spectral invariants are also homotopy invariants, these results provide information about the invariants of these operators. Categories:58J70, 43A77

17. CJM 2004 (vol 56 pp. 638)

Śniatycki, Jędrzej
 Multisymplectic Reduction for Proper Actions We consider symmetries of the Dedonder equation arising from variational problems with partial derivatives. Assuming a proper action of the symmetry group, we identify a set of reduced equations on an open dense subset of the domain of definition of the fields under consideration. By continuity, the Dedonder equation is satisfied whenever the reduced equations are satisfied. Keywords:Dedonder equation, multisymplectic structure, reduction,, symmetries, variational problemsCategories:58J70, 35A30

18. CJM 2004 (vol 56 pp. 590)

Ni, Yilong
 The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary We study the Riemannian Laplace-Beltrami operator $L$ on a Riemannian manifold with Heisenberg group $H_1$ as boundary. We calculate the heat kernel and Green's function for $L$, and give global and small time estimates of the heat kernel. A class of hypersurfaces in this manifold can be regarded as approximations of $H_1$. We also restrict $L$ to each hypersurface and calculate the corresponding heat kernel and Green's function. We will see that the heat kernel and Green's function converge to the heat kernel and Green's function on the boundary. Categories:35H20, 58J99, 53C17

19. CJM 2002 (vol 54 pp. 1086)

Polterovich, Iosif
 Combinatorics of the Heat Trace on Spheres We present a concise explicit expression for the heat trace coefficients of spheres. Our formulas yield certain combinatorial identities which are proved following ideas of D.~Zeilberger. In particular, these identities allow to recover in a surprising way some known formulas for the heat trace asymptotics. Our approach is based on a method for computation of heat invariants developed in [P]. Categories:05A19, 58J35

20. CJM 2000 (vol 52 pp. 695)

Carey, A.; Farber, M.; Mathai, V.
 Correspondences, von Neumann Algebras and Holomorphic $L^2$ Torsion Given a holomorphic Hilbertian bundle on a compact complex manifold, we introduce the notion of holomorphic $L^2$ torsion, which lies in the determinant line of the twisted $L^2$ Dolbeault cohomology and represents a volume element there. Here we utilise the theory of determinant lines of Hilbertian modules over finite von~Neumann algebras as developed in \cite{CFM}. This specialises to the Ray-Singer-Quillen holomorphic torsion in the finite dimensional case. We compute a metric variation formula for the holomorphic $L^2$ torsion, which shows that it is {\it not\/} in general independent of the choice of Hermitian metrics on the complex manifold and on the holomorphic Hilbertian bundle, which are needed to define it. We therefore initiate the theory of correspondences of determinant lines, that enables us to define a relative holomorphic $L^2$ torsion for a pair of flat Hilbertian bundles, which we prove is independent of the choice of Hermitian metrics on the complex manifold and on the flat Hilbertian bundles. Keywords:holomorphic $L^2$ torsion, correspondences, local index theorem, almost KÃ¤hler manifolds, von~Neumann algebras, determinant linesCategories:58J52, 58J35, 58J20
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