Willmore and Generalized Willmore energies in space forms

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2016-08

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Abstract

This PhD dissertation consists of two sections. In the first part of this work, we study a Generalized Willmore flow equation and its numerical applications. First, we derive the time dependent equation which relates to the geometric evolution of a Generalized Willmore flow on a general immersion. Then, we transform this equation into a coupled system of second order nonlinear PDEs together with the mean curvature formula on a general immersion in R3. The important point is that the Gauss curvature appears in terms of the second order derivatives in our coupled system. Moreover, we study finite element numerical solutions for steady-state cases obtained with the help of the FEMuS (Finite Element Multiphysics Solver) library. We use automatic differentiation (AD) tools to compute the exact Jacobian of the coupled PDE system subject to Dirichlet boundary conditions. Next, we study the Generalized Willmore flow in the graph case. This equation is reformulated in divergence form as a coupled system of second order nonlinear PDEs together with the mean curvature formula for the graph case. In fact, the Gauss curvature does not appear in our coupled system for the graph case. Furthermore, we study finite element numerical solutions for steady-state cases and use AD tools to compute the exact Jacobian of the coupled PDE system subject to different boundary conditions such as Dirichlet and Dirichlet-Neumann boundary conditions. This is a novel method - never used previously on computing/constructing Willmore surfaces. We study the accuracy of the algorithm by providing nontrivial steady-state numerical solutions to compute the Jacobian in the Newton linearization of the finite element weak formulation.

In the second part, we study Willmore-type energies and Willmore-type immersions in space forms. In fact, we introduce the notion of deformed Willmore energy for a space form M3(k0) as W¯=∫M(H2+k1)dS such that the constants k0, k1 are independent. Next, we discuss the corresponding Euler-Lagrange equation for the deformed energy. Its novel contributions include the following: \begin{itemize} \item deriving the Euler-Lagrange equation corresponding to the classical Willmore functional if we consider surfaces immersed in a general ambient space form M3(k0) of constant sectional curvature k0. In addition, we deduce the Euler-Lagrange equation of the deformed Willmore energy in a space form, in a unified way, using an extrinsic Laplace-Beltrami operator (which depends on both the surface metric, and the ambient space form). We consider both the case of closed surfaces and the one of surfaces with boundary, for which we gave and discussed the necessary boundary value conditions, which the previous literature failed to do;

\item determining the natural choices of the constant k1 in the deformed Willmore functional such that the corresponding the Euler-Lagrange equation which is given by

Unknown environment 'itemize'\Delta_{g}H+2H(H^2-K)=0.$$ This is the same form that is obtained for the classical Willmore functional in the Euclidean ambient space. \end{itemize} Thus, this dissertation provides a bridge between prior works in the field, as well as a novel approach both from a purely mathematical point of view and in the context of applications.\Delta_{g}H+2H(H^2-K)=0.$$ This is the same form that is obtained for the classical Willmore functional in the Euclidean ambient space. \end{itemize} Thus, this dissertation provides a bridge between prior works in the field, as well as a novel approach both from a purely mathematical point of view and in the context of applications.

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Willmore Energy

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