The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities

We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in L_p-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points. We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points. We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.     

Resolvents of cone pseudodifferential operators,asymptotic expansions and applications

We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula.      

The Atiyah–Bott–Lefschetz Theorem for Manifolds with Conical Singularities

We establish an Atiyah–Bott–Lefschetz formula for elliptic operators on manifolds with conical singular points.     

Local solvability for semilinear Fuchsian equations

We consider a semilinear elliptic operator P on a manifold B with a conical singular point. We assume P is Fuchs type in the linear part and has a non–linear lower order therms. Using the Schauder fixed point theorem, we prove the local solvability of P near the conical point in the weighted Sobolev spaces.     

The spectral function of a Riemannian orbifold

Hörmander’s theorem on the asymptotics of the spectral function of an elliptic operator is extended to the setting of compact Riemannian orbifolds. In contrast to the manifold case, the asymptotics depend on the isotropy type of the point at which the spectral function is computed. It is shown that “on average” the eigenfunctions of the operator are larger at singular points than at manifold points, by a factor of the order of the isotropy type. A sketch of a more direct approach to the wave trace formula on orbifolds is also given, obtaining results already shown separately by M. Sandoval and Y. Kordyukov in the setting of Riemmannian foliations.     

A sampling theorem on homogeneous manifolds

We consider a generalization of entire functions of spherical exponential type and Lagrangian splines on manifolds. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a manifold is uniquely determined by its values on some discrete sets of points.

The main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using Lagrangian splines.

     

A relative morse index for the symplectic action

The notion of a Morse index of a function on a finite-dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold. In this paper, we define for any pair of critical points of a a relative Morse index, which corresponds to the difference of the two Morse indices in finite dimensions. It is based on the spectral flow of the Hessian of a and can be identified with a topological invariant recently defined by Viterbo, and with the dimension of the space of trajectories between the two critical points.     

Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations

In this paper, the authors study the asymptotically linear elliptic equation on manifold with conical singularities ??_Bu + λu = a(z)f(u), u ≥ 0 in R^N₊,where N = n + 1 ≥ 3, λ > 0, z = t,x₁,· · · ,x_n, and ?_B = (t?_t)^{2 + ?2_x1 + · · · + ?2_xn. Combining properties of cone-degenerate operator, the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation, we obtain a positive solution under some suitable conditions on a and f.}     

Realizations of Differential Operators on Conic Manifolds with Boundary

We study the closed extensions (realizations) of differential operators subject to homogeneous boundary conditions on weighted L _p -Sobolev spaces over a manifold with boundary and conical singularities. Under natural ellipticity conditions we determine the domains of the minimal and the maximal extension. We show that both are Fredholm operators and give a formula for the relative index. Mathematics Subject Classifications (2000): Primary 58J32; Secondary 35G70, 35S15.     

On a formula for the spectral flow and its applications

We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi‐Riemannian geodesic, and we compute its value in terms of the Maslov index (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Index Formulae for Pseudodifferential Operators with Discontinuous Symbols

A class of zero order pseudodifferential operators on a closed manifold is considered, with symbols admitting a first kind discontinuity at a codimension one submanifold. A condition is found for such operators to be Fredholm. The formula for the index of such operators is derived, expressed in the topological terms. This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotics of the parabolic Green function for an elliptic operator on a manifold with conical points

We construct the asymptotics ast→0 of the trace of the operator exp(?tP) for an elliptic operatorP on a manifold with conical points.     

The index of elliptic operators on manifolds with conical points

For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the "eta" invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be nonzero. Moreover, we introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.     

The local index formula in noncommutative geometry

In noncommutative geometry a geometric space is described from a spectral vantage point, as a tripleA, H, D consisting of a *-algebraA represented in a Hilbert spaceH together with an unbounded selfadjoint operatorD, with compact resolvent, which interacts with the algebra in a bounded fashion. This paper contributes to the advancement of this point of view in two significant ways: (1) by showing that any pseudogroup of transformations of a manifold gives rise to such a spectral triple of finite summability degree, and (2) by proving a general, in some sense universal, local index formula for arbitrary spectral triples of finite summability degree, in terms of the Dixmier trace and its residue-type extension.We dedicate this paper to Misha Gromov     

Elliptic Operators on Manifolds with Singularities and K-Homology

Elliptic operators on smooth compact manifolds are classified by K-homology. We prove that a similar classification is valid also for manifolds with simplest singularities: isolated conical points and edges. The main ingredients of the proof of these results are: Atiyah–Singer difference construction in the noncommutative case and Poincaré isomorphism in K-theory for (our) singular manifolds. As an application we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with edges.Mathematics Subject Classification (2000): 58J05(Primary), 19K33 35S35 47L15(Secondary)(Received: June 2004)     

Boundary Problems for Dirac-Type Operators on Manifolds with Multi-Cylindrical End Boundaries

The goal of this paper is to establish a geometric program to study elliptic pseudodifferential boundary problems which arise naturally under cutting and pasting of geometric and spectral invariants of Dirac-type operators on manifolds with corners endowed with multi-cylindrical, or b-type, metrics and ‘b-admissible’ partitioning hypersurfaces. We show that the Cauchy data space of a Dirac operator on such a manifold is Lagrangian for the self-adjoint case, the corresponding Calderón projector is a b-pseudodifferential operator of order 0, characterize Fredholmness, prove relative index formulæ, and solve the Bojarski conjecture. Mathematics Subject Classifications (2000): 58J28, 58J52.     

Le saut en zéro de la fonction de décalage spectral

We give a formula for the jump at zero of the spectral shift function associated with Schrödinger operator on manifolds with conical ends. We show that, according to its decay, a zero energy resonant state has a non-integer contribution.     

Pontrjagin numbers and pseudo-riemannian geometry

Summary An integral formula for the Pontrjagin numbers of a compact orientable real 4k dimensional differentiable manifold which has a pseudo-Riemannian metric is derived. This formula allows the Pontrjagin numbers to be expressed in terms of the index, or signature, of the differentiable manifold. The application of these formulae to the four dimensional Lorentzian manifolds of the general theory of relativity is discussed. A corresponding formula for the Chern numbers of a complex differentiable manifold with a Hermitian metric is also given.     

单位球上线丛的不变Cauchy-Riemann算子

L_m,E是Kahler流形M上Hermite丛E第m阶Cauchy－Riemann算子,给定一定条件,L_m,E是L_1,E的多项式．当考虑M是黎曼面时,得到公式(16)．当E为Bⁿ的典范线丛,证明了L_m,E=(?)(L_1,E+(j-1)(j-2)).     

A generalization of the Leray—Schauder index formula

This paper generalizes the Leray-Schauder index formula to the case where the inverse image of a point consists of a smooth manifold, assuming some nondegeneracy condition is satisfied on the manifold. The result states that the index is the Euler characteristic of a certain vector bundle over the manifold. Under slightly stronger nondegeneracy conditions, the index is in fact the Euler characteristic of the manifold. The paper also includes a discussion of the Euler characteristic for vector bundles and a simple proof of the Gauss-Bonnet-Chern theorem.