On the Mean Curvature Flow of Submanifolds in the Standard Gaussian Space
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Results in Mathematics
On the Mean Curvature Flow of Submanifolds in the Standard Gaussian Space An-Min Li, Xingxiao Li, and Di Zhang Abstract. In this paper, we study the regular geometric behavior of the mean curvature flow (MCF) of submanifolds in the standard Gauss2 ian metric space (Rm+p , e−|x| /m g) where (Rm+p , g) is the standard Eum+p denotes the position vector. Note that, as a clidean space and x ∈ R 2 special Riemannian manifold, (Rm+p , e−|x| /m g) has an unbounded curvature. Up to a family of diffeomorphisms on M m , the mean curvature flow we considered here turns out to be equivalent to a special variation of the “conformal mean curvature flow” which we have introduced previously. The main theorem of this paper indicates, geometrically, that any immersed compact submanifold in the standard Gaussian space, with the square norm of the position vector being not equal to m, will blow up at a finite time under the mean curvature flow, in the sense that either the position or the curvature blows up to infinity; Moreover, by this main theorem, the interval [0, T ) of time in which the flowing submanifolds keep regular has some certain optimal upper bound, and it can reach the bound if and only if the initial submanifold either shrinks to the origin or expands uniformly to infinity under the flow. Besides the main theorem, we also obtain some other interesting conclusions which not only play their key roles in proving the main theorem but also characterize in part the geometric behavior of the flow, being of independent significance. Mathematics Subject Classification. Primary 53B44; Secondary 53B40. Keywords. Mean curvature flow, Gaussian space, blow-up of the curvature.
Research supported by National Natural Science Foundation of China (No. 11631002, No. 11821001, No. 11890663, No. 11961131001, No. 11671121, No. 11871197 and No. 11971153). 0123456789().: V,-vol
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Contents 1. Introduction 2. The Short-Time Existence and the Uniqueness of the Solution 3. Basic Evolution Formulae 4. Some of the Key Results Used for the Main Theorem 5. Higher-Order Derivative Estimates and the Blow-Up Argument 6. Proof of the Main Theorem References
1. Introduction Let M be a compact manifold of dimension m ≥ 2, and (Rm+p , g) (p ≥ 1) the Euclidean space with the standard flat metric g. Then the Riemannian 2 manifold (Rm+p , e−|x| /m g), where |x| denotes the standard norm of the position vector x ∈ Rm+p , is exactly the standard Gaussian metric space. Thus, for any immersion x : M → Rm+p , we have two induced metrics g := x∗ g, 2 1 g˜ = e− m |x| g and, accordingly, we also have two mean curvature vectors H, ˜ A simple computation shows that these two mean curvature vectors are H. related to each other by ˜ = e m1 |x|2 (H + x⊥ ). H
(1.1) 2
As is well known, the Gaussian space (Rm+p , e−|x| /m g) plays important roles both in physics and in mathematics; In particular, as a special Riemannian manifold, it has a typical geometric structure of non-constant curvature. 2 Moreover, while the
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