This thesis is twofold. In the first part, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given and a new curvature condition on abstract metric measure spaces is defined. In the second part of the thesis a proof of the identification of the q-heat equation with the gradient flow of the Renyi (3-p)-Renyi entropy functional in the p-Wasserstein space is given. For that, a further study of the q-heat flow is presented including a condition for its mass preservation.

Any Riemannian manifold possesses a minimal solution of the heat equation for the Dirichlet Laplacian, called the heat kernel. During the last decades many authors investigated geometric properties of the manifold such that its heat kernel fulfills a so-called Gaussian upper bound. Especially compact and non-compact manifolds with lower bounded Ricci curvature have been examined and provide such Gaussian estimates. In the compact case it ended even with integral Ricci curvature assumptions. The important techniques to obtain Gaussian bounds are the symmetrization procedure for compact manifolds and relative Faber-Krahn estimates or gradient estimates for the heat equation, where the first two base on isoperimetric properties of certain sets. In this thesis, we generalize the existing results to the following. Locally uniform integral bounds on the negative part of Ricci curvature lead to Gaussian upper bounds for the heat kernel, no matter whether the manifold is compact or not. Therefore, we show local isoperimetric inequalities under this condition and use relative Faber-Krahn estimates to derive explicit Gaussian upper bounds. If the manifold is compact, we can even generalize the integral curvature condition to the case that the negative part of Ricci curvature is in the so-called Kato class. We even obtain uniform Gaussian upper bounds using gradient estimate techniques. Apart from the geometric generalizations for obtaining Gaussian upper bounds we use those estimates to generalize Bochner’s theorem. More precisely, the estimates for the heat kernel obtained above lead to ultracontractive estimates for the heat semigroup and the semigroup generated by the Hodge Laplacian. In turn, we can formulate rigidity results for the triviality of the first cohomology group if the amount of curvature going below a certain positive threshold is small in a suitable sense. If we can only assume such smallness of the negative part of the Ricci curvature, we can bound the Betti number by explicit terms depending on the generalized curvature assumptions in a uniform manner, generalizing certain existing results from the cited literature. / Jede Riemannsche Mannigfaltigkeit besitzt eine minimale Lösung für die Wärmeleitungsgleichung des zur Mannigfaltigkeit gehörigen Dirichlet-Laplaceoperators, den Wärmeleitungskern. Während der letzten Jahrzehnte fanden viele Autoren geometrische Eigenschaften der Mannigfaltigkeiten unter welchen der Wärmeleitungskern eine sogenannte Gaußsche obere Abschätzung besitzt. Insbesondere bestizen sowohl kompakte als auch nichtkompakte Mannigfaltigkeiten mit nach unten beschränkter Ricci-Krümmung solche Gaußschen Abschätzungen. Im kompakten Fall reichten bisher sogar Integralbedingungen an die Ricci-Krümmung aus. Die wichtigen Techniken, um Gaußsche Abschätzungen zu erhalten, sind die Symmetrisierung für kompakte Mannigfaltigkeiten und relative Faber-Krahn- und Gradientenabschätzungen für die Wärmeleitungsgleichung, wobei die ersten beiden auf isoperimetrischen Eigenschaften gewisser Mengen beruhen. In dieser Arbeit verallgemeinern wir die bestehenden Resultate im folgenden Sinne. Lokal gleichmäßig beschränkte Integralschranken an den Negativteil der Ricci-Krümmung ergeben Gaußsche obere Abschätzungen sowohl im kompakten als auch nichtkompakten Fall. Dafür zeigen wir lokale isoperimetrische Ungleichungen unter dieser Voraussetzung und nutzen die relativen Faber-Krahn-Abschätzungen für eine explizite Gaußsche Schranke. Für kompakte Mannigfaltigkeiten können wir sogar die Integralschranken an den Negativteil der Ricci-Krümmung durch die sogenannte Kato-Bedingung ersetzen. In diesem Fall erhalten wir gleichmäßige Gaußsche Abschätzungen mit einer Gradientenabschätzung. Neben den geometrischen Verallgemeinerungen für Gaußsche Schranken nutzen wir unsere Ergebnisse, um Bochners Theorem zu verallgemeinern. Wärmeleitungskernabschätzungen ergeben ultrakontraktive Schranken für die Wärmeleitungshalbgruppe und die Halbgruppe, die durch den Hodge-Operator erzeugt wird. Damit können wir Starrheitseigenschaften für die erste Kohomologiegruppe zeigen, wenn der Teil der Ricci-Krümmung, welcher unter einem positiven Level liegt, in einem bestimmten Sinne klein genug ist. Wenn der Negativteil der Ricci-Krümmung nicht zu groß ist, können wir die erste Betti-Zahl noch immer explizit uniform abschätzen.

This thesis is twofold. In the first part, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given and a new curvature condition on abstract metric measure spaces is defined. In the second part of the thesis a proof of the identification of the q-heat equation with the gradient flow of the Renyi (3-p)-Renyi entropy functional in the p-Wasserstein space is given. For that, a further study of the q-heat flow is presented including a condition for its mass preservation.

Any Riemannian manifold possesses a minimal solution of the heat equation for the Dirichlet Laplacian, called the heat kernel. During the last decades many authors investigated geometric properties of the manifold such that its heat kernel fulfills a so-called Gaussian upper bound. Especially compact and non-compact manifolds with lower bounded Ricci curvature have been examined and provide such Gaussian estimates. In the compact case it ended even with integral Ricci curvature assumptions. The important techniques to obtain Gaussian bounds are the symmetrization procedure for compact manifolds and relative Faber-Krahn estimates or gradient estimates for the heat equation, where the first two base on isoperimetric properties of certain sets. In this thesis, we generalize the existing results to the following. Locally uniform integral bounds on the negative part of Ricci curvature lead to Gaussian upper bounds for the heat kernel, no matter whether the manifold is compact or not. Therefore, we show local isoperimetric inequalities under this condition and use relative Faber-Krahn estimates to derive explicit Gaussian upper bounds. If the manifold is compact, we can even generalize the integral curvature condition to the case that the negative part of Ricci curvature is in the so-called Kato class. We even obtain uniform Gaussian upper bounds using gradient estimate techniques. Apart from the geometric generalizations for obtaining Gaussian upper bounds we use those estimates to generalize Bochner’s theorem. More precisely, the estimates for the heat kernel obtained above lead to ultracontractive estimates for the heat semigroup and the semigroup generated by the Hodge Laplacian. In turn, we can formulate rigidity results for the triviality of the first cohomology group if the amount of curvature going below a certain positive threshold is small in a suitable sense. If we can only assume such smallness of the negative part of the Ricci curvature, we can bound the Betti number by explicit terms depending on the generalized curvature assumptions in a uniform manner, generalizing certain existing results from the cited literature. / Jede Riemannsche Mannigfaltigkeit besitzt eine minimale Lösung für die Wärmeleitungsgleichung des zur Mannigfaltigkeit gehörigen Dirichlet-Laplaceoperators, den Wärmeleitungskern. Während der letzten Jahrzehnte fanden viele Autoren geometrische Eigenschaften der Mannigfaltigkeiten unter welchen der Wärmeleitungskern eine sogenannte Gaußsche obere Abschätzung besitzt. Insbesondere bestizen sowohl kompakte als auch nichtkompakte Mannigfaltigkeiten mit nach unten beschränkter Ricci-Krümmung solche Gaußschen Abschätzungen. Im kompakten Fall reichten bisher sogar Integralbedingungen an die Ricci-Krümmung aus. Die wichtigen Techniken, um Gaußsche Abschätzungen zu erhalten, sind die Symmetrisierung für kompakte Mannigfaltigkeiten und relative Faber-Krahn- und Gradientenabschätzungen für die Wärmeleitungsgleichung, wobei die ersten beiden auf isoperimetrischen Eigenschaften gewisser Mengen beruhen. In dieser Arbeit verallgemeinern wir die bestehenden Resultate im folgenden Sinne. Lokal gleichmäßig beschränkte Integralschranken an den Negativteil der Ricci-Krümmung ergeben Gaußsche obere Abschätzungen sowohl im kompakten als auch nichtkompakten Fall. Dafür zeigen wir lokale isoperimetrische Ungleichungen unter dieser Voraussetzung und nutzen die relativen Faber-Krahn-Abschätzungen für eine explizite Gaußsche Schranke. Für kompakte Mannigfaltigkeiten können wir sogar die Integralschranken an den Negativteil der Ricci-Krümmung durch die sogenannte Kato-Bedingung ersetzen. In diesem Fall erhalten wir gleichmäßige Gaußsche Abschätzungen mit einer Gradientenabschätzung. Neben den geometrischen Verallgemeinerungen für Gaußsche Schranken nutzen wir unsere Ergebnisse, um Bochners Theorem zu verallgemeinern. Wärmeleitungskernabschätzungen ergeben ultrakontraktive Schranken für die Wärmeleitungshalbgruppe und die Halbgruppe, die durch den Hodge-Operator erzeugt wird. Damit können wir Starrheitseigenschaften für die erste Kohomologiegruppe zeigen, wenn der Teil der Ricci-Krümmung, welcher unter einem positiven Level liegt, in einem bestimmten Sinne klein genug ist. Wenn der Negativteil der Ricci-Krümmung nicht zu groß ist, können wir die erste Betti-Zahl noch immer explizit uniform abschätzen.

The interaction between the study of geometric and analytic aspects of Riemannian manifolds and that of graphs is a very amazing subject. The study of synthetic curvature notions on graphs adds new contributions to this topic. In this thesis, we mainly study two kinds of synthetic curvature notions: the Ollivier-Ricci cuvature on locally finite graphs and the combinatorial curvature on infinite semiplanar graphs. In the first part, we study the Ollivier-Ricci curvature. As known in Riemannian geometry, a lower Ricci curvature bound prevents geodesics from diverging too fast on average. We translate this Riemannian idea into a combinatorial setting using the Olliver-Ricci curvature notion. Note that on a graph, the analogue of geodesics starting in different directions, but eventually approaching each other again, would be a triangle. We derive lower and upper Ollivier-Ricci curvature bounds on graphs in terms of number of triangles, which is sharp for instance for complete graphs. We then describe the relation between Ollivier-Ricci curvature and the local clustering coefficient, which is an important concept in network analysis introduced by Watts-Strogatz. Furthermore, positive lower boundedness of Ollivier-Ricci curvature for neighboring vertices imply the existence of at least one triangle. It turns out that the existence of triangles can also improve Lin-Yau\'s curvature dimension inequality on graphs and then produce an implication from Ollivier-Ricci curvature lower boundedness to the curvature dimension inequality. The existence of triangles prevents a graph from being bipartite. A finite graph is bipartite if and only if its largest eigenvalue equals 2. Therefore it is natural that Ollivier-Ricci curvature is closely related to the largest eigenvalue estimates. We combine Ollivier-Ricci curvature notion with the neighborhood graph method developed by Bauer-Jost to study the spectrum estimates of a finite graph. We can always obtain nontrivial estimates on a non-bipartite graph even if its curvature is nonpositive. This answers one of Ollivier\'s open problem in the finite graph setting. In the second part of this thesis, we study systematically infinite semiplanar graphs with nonnegative combinatorial curvature. Unlike the previous Gauss-Bonnet formula approach, we explore an Alexandrov approach based on the observation that the nonnegative combinatorial curvature on a semiplanar graph is equivalent to nonnegative Alexandrov curvature on the surface obtained by replacing each face by a regular polygon of side length one with the same facial degree and gluing the polygons along common edges. Applying Cheeger-Gromoll splitting theorem on the surface, we give a metric classification of infinite semiplanar graphs with nonnegative curvature. We also construct the graphs embedded into the projective plane minus one point. Those constructions answer a question proposed by Chen. We further prove the volume doubling property and Poincare inequality which make the running of Nash-Moser iteration possible. We in particular explore the volume growth behavior on Archimedean tilings on a plane and prove that they satisfy a weak version of relative volume comparison with constant 1. With the above two basic inequalities in hand, we study the geometric function theory of infinite semiplanar graphs with nonnegative curvature. We obtain the Liouville type theorem for positive harmonic functions, the parabolicity. We also prove a dimension estimate for polynomial growth harmonic functions, which is an extension of the solution of Colding-Minicozzi of a conjecture of Yau in Riemannian geometry.

The interaction between the study of geometric and analytic aspects of Riemannian manifolds and that of graphs is a very amazing subject. The study of synthetic curvature notions on graphs adds new contributions to this topic. In this thesis, we mainly study two kinds of synthetic curvature notions: the Ollivier-Ricci cuvature on locally finite graphs and the combinatorial curvature on infinite semiplanar graphs. In the first part, we study the Ollivier-Ricci curvature. As known in Riemannian geometry, a lower Ricci curvature bound prevents geodesics from diverging too fast on average. We translate this Riemannian idea into a combinatorial setting using the Olliver-Ricci curvature notion. Note that on a graph, the analogue of geodesics starting in different directions, but eventually approaching each other again, would be a triangle. We derive lower and upper Ollivier-Ricci curvature bounds on graphs in terms of number of triangles, which is sharp for instance for complete graphs. We then describe the relation between Ollivier-Ricci curvature and the local clustering coefficient, which is an important concept in network analysis introduced by Watts-Strogatz. Furthermore, positive lower boundedness of Ollivier-Ricci curvature for neighboring vertices imply the existence of at least one triangle. It turns out that the existence of triangles can also improve Lin-Yau\''s curvature dimension inequality on graphs and then produce an implication from Ollivier-Ricci curvature lower boundedness to the curvature dimension inequality. The existence of triangles prevents a graph from being bipartite. A finite graph is bipartite if and only if its largest eigenvalue equals 2. Therefore it is natural that Ollivier-Ricci curvature is closely related to the largest eigenvalue estimates. We combine Ollivier-Ricci curvature notion with the neighborhood graph method developed by Bauer-Jost to study the spectrum estimates of a finite graph. We can always obtain nontrivial estimates on a non-bipartite graph even if its curvature is nonpositive. This answers one of Ollivier\''s open problem in the finite graph setting. In the second part of this thesis, we study systematically infinite semiplanar graphs with nonnegative combinatorial curvature. Unlike the previous Gauss-Bonnet formula approach, we explore an Alexandrov approach based on the observation that the nonnegative combinatorial curvature on a semiplanar graph is equivalent to nonnegative Alexandrov curvature on the surface obtained by replacing each face by a regular polygon of side length one with the same facial degree and gluing the polygons along common edges. Applying Cheeger-Gromoll splitting theorem on the surface, we give a metric classification of infinite semiplanar graphs with nonnegative curvature. We also construct the graphs embedded into the projective plane minus one point. Those constructions answer a question proposed by Chen. We further prove the volume doubling property and Poincare inequality which make the running of Nash-Moser iteration possible. We in particular explore the volume growth behavior on Archimedean tilings on a plane and prove that they satisfy a weak version of relative volume comparison with constant 1. With the above two basic inequalities in hand, we study the geometric function theory of infinite semiplanar graphs with nonnegative curvature. We obtain the Liouville type theorem for positive harmonic functions, the parabolicity. We also prove a dimension estimate for polynomial growth harmonic functions, which is an extension of the solution of Colding-Minicozzi of a conjecture of Yau in Riemannian geometry.