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Jahresbericht der Deutschen Mathematiker-Vereinigung

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12-12-2016 | Survey Article

Variational Methods in Geometry

Author: Michael Struwe

Published in: Jahresbericht der Deutschen Mathematiker-Vereinigung | Issue 2/2017

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Abstract

Variational principles are ubiquitous in nature. Many geometric objects such as geodesics or minimal surfaces allow variational characterizations. We recall some basic ideas in the calculus of variations, also relevant for some of the most advanced research in the field today, and show how a subtle variation of standard methods can lead to surprising improvements, with numerous applications.

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Jahresbericht der Deutschen Mathematiker-Vereinigung

Der „Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)“ versteht sich als ein Schaufenster für Mathematik. In Übersichtsartikeln und Berichten aus der Forschung soll für möglichst viele LeserInnen verständlich und interessant über aktuelle und wichtige Entwicklungen der Mathematik berichtet werden.

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Here $H^{1}([0,1];\mathbb{R}^{n})$ is the Sobolev spaces of curves $\gamma \in L^{2}([0,1];\mathbb{R}^{n})$ with distributional derivative $\dot{\gamma }\in L^{2}([0,1];\mathbb{R} ^{n})$.

Without reference to Riemann normal coordinates this may be seen as follows: Suppose for simplicity that $S\subset \mathbb{R}^{n}$ is an oriented hypersurface with smooth unit normal vector field $\nu $, and let $\gamma_{0},\gamma_{1}\in H^{1}([0,1];\mathbb{R}^{n})$ be curves on $S$ with endpoints $\gamma_{0}(0)=\gamma_{1}(0)=p$, $\gamma_{0}(1)=\gamma_{1}(1)=q$ having Euclidean distance $|p-q|< \delta $, such that

$$ |\dot{\gamma }_{0}|^{2}\equiv E( \gamma_{0}) =E(\gamma_{1}) =\inf \bigl\{ E( \gamma );\ \gamma \in \varGamma_{p,q}\bigr\} < \delta^{2} $$

(7)

for some $\delta >0$. Suppose that $\gamma_{0}\neq \gamma_{1}$ and expand

$$ E(\gamma_{1})=E(\gamma_{0})+ 2 \int_{0}^{1}\dot{\gamma }_{0}\cdot ( \dot{ \gamma }_{1}-\dot{\gamma }_{0})\,dt + \int_{0}^{1}|\dot{\gamma } _{1}-\dot{\gamma }_{0}|^{2}\,dt. $$

(8)

Let $\nu_{0}=\nu \circ \gamma_{0}$. Using (6) and observing that orthogonality $\nu_{0}\perp T_{\gamma_{0}}S$ gives $\dot{\gamma }_{0} \cdot \nu_{0}\equiv 0$ and also allows to bound $|\nu_{0}\cdot (\gamma _{1}-\gamma_{0})|\le C|\gamma_{1}-\gamma_{0}|^{2}$, upon integrating by parts we find

$$ \begin{aligned} &- \int_{0}^{1}\dot{\gamma }_{0}\cdot (\dot{ \gamma }_{1}-\dot{\gamma } _{0})dt = \int_{0}^{1}\ddot{\gamma }_{0}\cdot ( \gamma_{1}-\gamma_{0})\,dt = \int_{0}^{1}\ddot{\gamma }_{0}\cdot \nu_{0}\ \nu_{0}\cdot (\gamma _{1}- \gamma_{0})\,dt \\ &\quad=- \int_{0}^{1}\bigl(\dot{\gamma }_{0}\cdot d \nu (\gamma_{0}) \dot{\gamma }_{0}\bigr) \nu_{0} \cdot (\gamma_{1}-\gamma_{0})\,dt \le C\sup_{t} \bigl(|\dot{\gamma }_{0}|^{2}|\gamma_{1}- \gamma_{0}|^{2}\bigr) \le C\delta ^{2} \int_{0}^{1}|\dot{\gamma }_{1}-\dot{\gamma }_{0}|^{2}\,dt, \end{aligned} $$

which in view of (8) contradicts (7) if $\delta >0$ is sufficiently small. Note that $\gamma_{1}-\gamma_{0}\in H_{0}^{1}([0,1]; \mathbb{R}^{n})$.

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Title: Variational Methods in Geometry
Author: Michael Struwe
Publication date: 12-12-2016
Publisher: Springer Berlin Heidelberg
Published in: Jahresbericht der Deutschen Mathematiker-Vereinigung / Issue 2/2017
Print ISSN: 0012-0456
Electronic ISSN: 1869-7135
DOI: https://doi.org/10.1365/s13291-016-0155-0

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