Positively Curved Surfaces in the Three-sphere
Positively Curved Surfaces in the Three-sphere
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发表时间：
2003-04
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通讯作者：
B. Andrews
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作者：
B. Andrews
在这次演讲中，我将讨论一个使用完全非线性抛物流来证明几何结果的例子。我要强调的事实是，有各种各样的几何抛物方程可供选择，为了得到最好的结果，选择最好的流是非常重要的。我将在三维球面的表面设置中说明这一点。对于球面上满足各种曲率方程的曲面，包括全脐形曲面、极小曲面和常平均曲率曲面以及本质平面，已有相当多的相关结果。抛物线流可以通过允许满足曲率不等式而不是等式的曲面来加强这种结果：这是由Huisken首先完成的，他使用平均曲率流将某些曲面变形为完全的脐形曲面。这引发了一个问题：“这种方法的最佳结果是什么？”——也就是说，定义一类曲面的最弱点曲率条件是什么，这些曲面可以缩回到大球体空间？从这些例子中可以猜出这个问题的答案。为了证明它，需要一个令人惊讶的进化方程的选择，这是由点曲率条件被保留的要求所迫的。最后，我将提到其他一些几何情形，在这些情形中，可以通过选择可能的最佳演化方程来证明强结果数学学科分类：53C44、53C40。关键词：曲面，曲率，抛物线方程。
AbstractIn this talk I will discuss an example of the use of fully nonlinear parabolicflows to prove geometric results. I will emphasise the fact that there is a widevariety of geometric parabolic equations to choose from, and to get the bestresults it can be very important to choose the best flow. I will illustrate thisin the setting of surfaces in a three-dimensional sphere.There are quite a few relevant results for surfaces in the sphere satisfy-ing various kinds of curvature equations, including totally umbillic surfaces,minimal surfaces and constant mean curvature surfaces, and intrinsically flatsurfaces. Parabolic flows can strengthen such results by allowing classes ofsurfaces satisfying curvature inequalities rather than equalities: This was firstdone by Huisken, who used mean curvature flow to deform certain classes ofsurfaces to totally umbillic surfaces. This motivates the question “What is theoptimal result of this kind?” — that is, what is the weakest pointwise curva-ture condition which defines a class of surfaces which retracts to the space ofgreat spheres?The answer to this question can be guessed in view of the examples. Toprove it requires a surprising choice of evolution equation, forced by the re-quirement that the pointwise curvature condition be preserved.I will conclude by mentioning some other geometric situations in whichstrong results can be proved by choosing the best possible evolution equation.2000 Mathematics Subject Classification: 53C44, 53C40.Keywords and Phrases: Surfaces, Curvature, Parabolic equations.