We provide geometric inequalities on $R^n$ and on general manifolds with nonnegative Ricci curvature by employing suitable monotone quantities along the flow of capacitary and $p$-capacitary potentials, as well as through related boundary value problems. Among the main achievements, we cite [(i)] a Willmore-type inequality on manifolds with nonnegative Ricci curvature leading in turn to the sharp Isoperimetric Inequality on $3$-manifolds with nonnegative Ricci curvature ; [(ii)] enhanced Kasue/Croke-Kleiner splitting theorems ; [(iii)] a generalised Minkowski-type inequality in $R^n$ holding with no assumptions on the boundary of the domain considered except for smoothness ; [(iv)] a complete discussion of maximal volume solutions to the least area problem with obstacle on Riemannian manifolds and its relation with the variational $p$-capacity.

In this work, turbulent premixed combustion was analyzed through CH (methylidyne) planar laser induced fluorescence (PLIF). Flame topography measurements in terms of flame surface density and curvature were calculated based on the flame front detected by the CH PLIF signal. The goal of this work was to investigate turbulent flames with extremely high turbulence intensity using a recently developed HiPilot burner (a Bunsen-type burner). The studies were first conducted on a series of piloted jet flames to validate the methodology, and then conducted on the highly turbulent flames generated by the HiPilot burner. All flames were controlled by combusting methane and air under a fuel to air equivalence ratio of Φ=1.05, and the Reynolds number varied from 7,385 to 28,360. Flame surface density fields and profiles for the HiPilot burner are presented. These flame surface density measurements showed an overall decrease with height above the burner. In addition, curvature statistics for the HiPilot flames were calculated and probability density functions of the curvature samples were determined. The probability density functions of curvature for the flames showed Gaussian-shaped distributions centered near zero curvature. To conclude, flame topography measurements were verified on jet flames and were demonstrated on the new HiPilot flames. / Master of Science / Optical diagnostics are powerful techniques that enable the study of turbulent flames without physical interruption. The optical diagnostic technique in this thesis implemented planar laser induced fluorescence. In planar laser induced fluorescence, a laser is used to excite a specific molecular species present within a two-dimensional plane in the flame. The excited species releases the extra energy by emission of light which is the signal captured on a camera. One useful purpose of using optical diagnostics, such as planar laser induced fluorescence, is the ability to image the flame structure of turbulent flames. The flame structure is significant for two reasons. First, the flame structure details how the chemistry of the flame interacts with the turbulent flow field. Second, the flame structure is directly related to the burning rate of the reactants. The primary contribution of this thesis investigated the two-dimensional flame structure of a newly designed burner named the HiPilot burner. However, in order to strengthen the fidelity of the methods for determining certain flame structure quantities a precursive analysis on the classical jet flame was completed. The results acquired show structural measurements of the HiPilot flames which contribute to the repository of data for the combustion community

Neste trabalho mostramos que hipersuperfícies completas da esfera Euclidiana S^4, com curvatura média constante e curvatura de Gauss-Kronecker nula são mínimas, sempre que o quadrado da norma da segunda forma fundamental for limitado superiormente. Além disso apresentamos uma descrisão local das hipersuperfícies mínimas e completas em S^5 com curvatura de Gauss- Kronecker nula e algumas hipóteses adicionais sobre as funções simétricas das curvaturas principais. / In this work we show that a complete hipersurface of the unitary sphere S^4, with constant mean curvature and zero Gauss-Kronecker curvature must be minimal, if the squared norm of the second fundamental form is bounded from above. Also, we present a local description for complete minimal hipersurfaces in S^5 with zero Gauss-Kronecker curvature, and some restrictions for the symmetric functions of the principal curvatures.

Neste trabalho mostramos que hipersuperfícies completas da esfera Euclidiana S^4, com curvatura média constante e curvatura de Gauss-Kronecker nula são mínimas, sempre que o quadrado da norma da segunda forma fundamental for limitado superiormente. Além disso apresentamos uma descrisão local das hipersuperfícies mínimas e completas em S^5 com curvatura de Gauss- Kronecker nula e algumas hipóteses adicionais sobre as funções simétricas das curvaturas principais. / In this work we show that a complete hipersurface of the unitary sphere S^4, with constant mean curvature and zero Gauss-Kronecker curvature must be minimal, if the squared norm of the second fundamental form is bounded from above. Also, we present a local description for complete minimal hipersurfaces in S^5 with zero Gauss-Kronecker curvature, and some restrictions for the symmetric functions of the principal curvatures.

<p>By applying a segmentation procedure to two different sets of computed tomography scans, two geometrical models of the abdominal aorta, containing one inlet and two outlets have been constructed. One of these depicts a healthy blood vessel while the other displays one afflicted with a Abdominal Aortic Aneurysm. </p><p>After inputting these geometries into the computational dynamics software FLUENT, six simulations of laminar, stationary flow of a fluid that was assumed to be Newtonian were performed. The mass flow rate across the model outlet boundaries was varied for the different simulations to produce a basis for a parameter analysis study. </p><p>The segmentation data was also used as input data to a surface description procedure which produced not only the surface itself, but also the first and second directional derivatives in every one of its defining spatial data points. These sets of derivatives were followingly applied in an additional procedure that calculated values of Gaussian curvature. </p><p>A parameter variance analysis was carried out to evaluate the performance of the surface generation procedure. An array of resultant surfaces and surface directional derivatives were obtained. Values of Gaussian curvature were calculated in the defining spatial data points of a few selected surfaces. </p><p>The curvature values of a selected data set were visualized through a contour plot as well as through a surface map. Comparisons between the curvature surface map and one wall shear stress surface map were made.</p>

By applying a segmentation procedure to two different sets of computed tomography scans, two geometrical models of the abdominal aorta, containing one inlet and two outlets have been constructed. One of these depicts a healthy blood vessel while the other displays one afflicted with a Abdominal Aortic Aneurysm. After inputting these geometries into the computational dynamics software FLUENT, six simulations of laminar, stationary flow of a fluid that was assumed to be Newtonian were performed. The mass flow rate across the model outlet boundaries was varied for the different simulations to produce a basis for a parameter analysis study. The segmentation data was also used as input data to a surface description procedure which produced not only the surface itself, but also the first and second directional derivatives in every one of its defining spatial data points. These sets of derivatives were followingly applied in an additional procedure that calculated values of Gaussian curvature. A parameter variance analysis was carried out to evaluate the performance of the surface generation procedure. An array of resultant surfaces and surface directional derivatives were obtained. Values of Gaussian curvature were calculated in the defining spatial data points of a few selected surfaces. The curvature values of a selected data set were visualized through a contour plot as well as through a surface map. Comparisons between the curvature surface map and one wall shear stress surface map were made.

Given a compact smooth manifold Mn without boundary and n ≥ 3, the Lp-norm of the curvature tensor, defines a Riemannian functional on the space of Riemannian metrics with unit volume M1. Consider C2,α-topology on M1 Rp remains invariant under the action of the group of diffeomorphisms D of M. So, Rp is defined on M1/ D. Our first result is that Rp restricted to the space M1/D has strict local minima at Riemannian metrics with constant sectional curvature for certain values of p. The product of spherical space forms and the product of compact hyperbolic manifolds are also critical point for Rp if they are product of same dimensional manifolds. We prove that these spaces are strict local minima for Rp restricted to M1/D. Compact locally symmetric isotropy irreducible metrics are critical points for Rp. We give a criteria for the local minima of Rp restricted to the conformal class of metrics of a given irreducible symmetric metric. We also prove that the metrics with constant bisectional curvature are strict local minima for Rp restricted to the space of Kahlar metrics with unite volume quotient by D. Next we consider the Riemannian functional given by In [GV], M. J. Gursky and J. A. Viaclovsky studied the local properties of the moduli space of critical metrics for the functional Ric2.We generalize their results for any p > 0.

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We discuss two topics in this talk. First we study compact Ricci-flat four dimensional manifolds without boundary and obtain point wise restrictions on curvature( not involving global quantities such as volume and diameter) which force the metric to be flat. We obtain the same conclusion for compact Ricci-flat K¨ahler surfaces with similar but weaker restrictions on holomorphic sectional curvature. Next we study the reaction ODE associated to the evolution of the Riemann curvature operator along the Ricci flow. We analyze the behavior of this ODE near algebraic curvature operators of certain special type that includes the Riemann curvature operators of various(locally) symmetric spaces. We explicitly show the existence of some solution curves to the ODE connecting the curvature operators of certain symmetric spaces. Although the results of these two themes are different, the underlying common feature is the reaction ODE which plays an important role in both.

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.