I will present the mean curvature flow in Euclidean spaces and the Lagrangian mean curvature flow. We will first study the mean curvature evolution of submanifolds in Euclidean spaces, with an emphasis on the case of hypersurfaces. Along the way we will demonstrate the basic techniques in the study of geometric flows in general (for example, various maximum principles and the treatment of singularities). After that we will move on to the study of Lagrangian mean curvature flows. We will make the relevant definitions and prove the fundamental result that the Lagrangian condition is preserved along the mean curvature flow in Kähler-Einstein manifolds, which started the extensive, and still ongoing, research on Lagrangian mean curvature flows. We will also define special Lagrangian submanifolds as calibrated submanifolds in Calabi-Yau manifolds. Finally, we will study the mean curvature flow of conormal bundles as submanifolds of C^n. Using some tools developed recently, we will show that if a surface has strictly negative curvatures, then away from the zero section, the Lagrangian mean curvature flow starting from a conormal bundle does not develop Type I singularities.

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

In this thesis we study singularity formation in two basic nonlinear equations in $n$ dimensions: nonlinear heat equation (also known as reaction-diffusion equation) and mean curvature flow equation. For the nonlinear heat equation, we show that for an important or natural open set of initial conditions the solution will blowup in finite time. We also characterize the blowup profile near blowup time. For the mean curvature flow we show that for an initial surface sufficiently close, in the Sobolev norm with the index greater than $\frac{n}{2} + 1$, to the standard n-dimensional sphere, the solution collapses in a finite time $t_*$, to a point. We also show that as $t\rightarrow t_*$, it looks like a sphere of radius $\sqrt{2n(t_*-t)}$.

I will present the mean curvature flow in Euclidean spaces and the Lagrangian mean curvature flow. We will first study the mean curvature evolution of submanifolds in Euclidean spaces, with an emphasis on the case of hypersurfaces. Along the way we will demonstrate the basic techniques in the study of geometric flows in general (for example, various maximum principles and the treatment of singularities). After that we will move on to the study of Lagrangian mean curvature flows. We will make the relevant definitions and prove the fundamental result that the Lagrangian condition is preserved along the mean curvature flow in Kähler-Einstein manifolds, which started the extensive, and still ongoing, research on Lagrangian mean curvature flows. We will also define special Lagrangian submanifolds as calibrated submanifolds in Calabi-Yau manifolds. Finally, we will study the mean curvature flow of conormal bundles as submanifolds of C^n. Using some tools developed recently, we will show that if a surface has strictly negative curvatures, then away from the zero section, the Lagrangian mean curvature flow starting from a conormal bundle does not develop Type I singularities.

We adress an obstacle problem for (graphical) mean curvature flow with Dirichlet boundary conditions. Using (an adapted form of) the standard implicit time-discretization scheme we derive the existence of distributional solutions satisfying an appropriate variational inequality. Uniqueness of this flow and asymptotic convergence towards the stationary solution is proven.

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.

In this thesis, we study the overfitting problem in supervised learning of classifiers from a geometric perspective. As with many inverse problems, learning a classification function from a given set of example-label pairs is an ill-posed problem, i.e., there exist infinitely many classification functions that can correctly predict the class labels for all training examples. Among them, according to Occam's razor, simpler functions are favored since they are less overfitted to training examples and are therefore expected to perform better on unseen examples. The standard technique to enforce Occam's razor is to introduce a regularization scheme, which penalizes some type of complexity of the learned classification function. Some widely used regularization techniques are functional norm-based (Tikhonov) techniques, ensemble-based techniques, early stopping techniques, etc. However, there is important geometric information in the learned classification function that is closely related to overfitting, and has been overlooked by previous methods. In this thesis, we study the complexity of a classification function from a new geometric perspective. In particular, we investigate the differential geometric structure in the submanifold corresponding to the estimator of the class probability P(y|x), based on the observation that overfitting produces rapid local oscillations and hence large mean curvature of this submanifold. We also show that our geometric perspective of supervised learning is naturally related to an elastic model in physics, where our complexity measure is a high dimensional extension of the surface energy in physics. This study leads to a new geometric regularization approach for supervised learning of classifiers. In our approach, the learning process can be viewed as a submanifold fitting problem that is solved by a mean curvature flow method. In particular, our approach finds the submanifold by iteratively fitting the training examples in a curvature or volume decreasing manner. Our technique is unified for both binary and multiclass classification, and can be applied to regularize any classification function that satisfies two requirements: firstly, an estimator of the class probability can be obtained; secondly, first and second derivatives of the class probability estimator can be calculated. For applications, where we apply our regularization technique to standard loss functions for classification, our RBF-based implementation compares favorably to widely used regularization methods for both binary and multiclass classification. We also design a specific algorithm to incorporate our regularization technique into the standard forward-backward training of deep neural networks. For theoretical analysis, we establish Bayes consistency for a specific loss function under some mild initialization assumptions. We also discuss the extension of our approach to situations where the input space is a submanifold, rather than a Euclidean space. / 2018-11-30T00:00:00Z

This thesis provides an analytical and two numerical methods for solving a parabolic equation of two-dimensional mean curvature flow with some applications. In analytical method, this equation is solved by Lie group analysis method, and in numerical method, two algorithms are implemented in MATLAB for solving this equation. A geometric algorithm and a step-wise algorithm; both are based on a deterministic game theoretic representation for parabolic partial differential equations, originally proposed in the genial work of Kohn-Serfaty [1]. / +46-767165881

Made available in DSpace on 2015-05-15T11:46:15Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 577287 bytes, checksum: 05b66c45fbee6c87496fa2601f5736ea (MD5) Previous issue date: 2013-05-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We study two types of Neumann problem related to Capillary problem and to the evolution of graphs under mean curvature flow in Riemannian manifolds endowed with a Killing vector field. In particular, we prove the existence of Killing graphs with prescribed mean curvature and prescribed boundary conditions. / Estudamos dois tipos de problemas relacionados com a Neumann problema capilar e à evolução dos gráficos sob fluxo de curvatura média em variedades Riemannianas dotados com um campo de vetores Killing. Em particular, provamos a existência de Matar gráficos prescrito com curvatura média e condições de contorno prescritas.

We adress an obstacle problem for (graphical) mean curvature flow with Dirichlet boundary conditions. Using (an adapted form of) the standard implicit time-discretization scheme we derive the existence of distributional solutions satisfying an appropriate variational inequality. Uniqueness of this flow and asymptotic convergence towards the stationary solution is proven.