In this work we study two problems about parabolic partial differential equations on Riemannian manifolds with conical singularities. The first problem we are concerned with is the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. By introducing so called weighted Hölder and Sobolev spaces with discrete asymptotics, we provide a complete existence and regularity theory for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. The second problem we study is the short time existence problem for the generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds, when the initial Lagrangian submanifold has isolated conical singularities that are modelled on stable special Lagrangian cones. First we use Lagrangian neighbourhood theorems for Lagrangian submanifolds with conical singularities to integrate the generalized Lagrangian mean curvature flow to a nonlinear parabolic equation of functions, and then, using the existence and regularity theory for the heat equation, we prove short time existence of the generalized Lagrangian mean curvature flow with isolated conical singularities by letting the conical singularities move around in the ambient space and the model cones to rotate by unitary transformations.

The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah-Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.

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

The structure of symplectic (canonical) transformations on manifolds with conical singularities is established. The operators associated with these transformations are defined in the weight spaces and their properties investigated.

The quantization of contact transformations of the cosphere bundle over a manifold with conical singularities is described. The index of Fredholm operators given by this quantization is calculated. The answer is given in terms of the Epstein-Melrose contact degree and the conormal symbol of the corresponding operator.

For general endomorphisms of elliptic complexes on manifolds with conical singularities, the semiclassical asymptotics of the Atiyah-Bott-Lefschetz number is calculated in terms of fixed points of the corresponding canonical transformation of the symplectic space.

Mixed elliptic problems are characterised by conditions that have a discontinuity on an interface of the boundary of codimension 1. The case of a smooth interface is treated in [3]; the investigation there refers to additional interface conditions and parametrices in standard Sobolev spaces. The present paper studies a necessary structure for the case of interfaces with conical singularities, namely, corner conormal symbols of such operators. These may be interpreted as families of mixed elliptic problems on a manifold with smooth interface. We mainly focus on second order operators and additional interface conditions that are holomorphic in an extra parameter. In particular, for the case of the Zaremba problem we explicitly obtain the number of potential conditions in this context. The inverses of conormal symbols are meromorphic families of pseudo-differential mixed problems referring to a smooth interface. Pointwise they can be computed along the lines [3].

We construct elliptic elements in the algebra of (classical pseudo-differential) operators on a manifold M with conical singularities. The ellipticity of any such operator A refers to a pair of principal symbols (σ0, σ1) where σ0 is the standard (degenerate) homogeneous principal symbol, and σ1 is the so-called conormal symbol, depending on the complex Mellin covariable z. The conormal symbol, responsible for the conical singularity, is operator-valued and acts in Sobolev spaces on the base X of the cone. The σ1-ellipticity is a bijectivity condition for all z of real part (n + 1)/2 − γ, n = dimX, for some weight γ. In general, we have to rule out a discrete set of exceptional weights that depends on A. We show that for every operator A which is elliptic with respect to σ0, and for any real weight γ there is a smoothing Mellin operator F in the cone algebra such that A + F is elliptic including σ1. Moreover, we apply the results to ellipticity and index of (operator-valued) edge symbols from the calculus on manifolds with edges.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18444号 / 理博第4004号 / 新制||理||1577(附属図書館) / 31322 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 加藤 毅, 教授 堤 誉志雄, 教授 小野 薫 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Estudamos a Torção Analítica para variedades com bordo e ainda com singuaridades do tipo cônico, mais especificamente, para um cone métrico limitado, com o propósito de investigar a extensão natural do Teorema de Cheeger Müller para tais espaços. Começamos determinando a Torção Analítica do disco e de variedades com o bordo totalmente geodésico, por meio de ferramentas geométricas desenvolvidas por J. Brüning e X. Ma. Posteriormente, usando ferramentas analíticas desenvolvidas por M. Spreafico, determinamos a Torção Analítica do cone sobre uma esfera de dimensão ímpar e provamos um teorema do tipo Cheeger Müller para este espaço. Mais ainda, provamos que o resualto de J. Brüning e X. Ma estende para o cone sobre uma esfera de dimensão ímpar / We study for Analytic Torsion of manifolds with boundary and also with conical singularities , more specifically, for a finite metric cone, with the purpose of investing the natural extension of the Cheeger Müller theorem for such spaces. we start by computing the Analytic Torsion of an any dimensional disc and of a manifold with totally boundary, by using geometric tools development by J. Brüning and X. Ma. Then, by using analytic tools development by M. Spreafico, we determine the Analytic Torsion of a cone over an odd dimensional sphere and we prove a theorem of Cheeger Müller type space. Moreover, we prove that the result of J. Brüning and X. Ma extends to the cone over an odd dimensional sphere