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Abstract The skew mean curvature flow is an evolution equation for a $$d$$ dimensional manifold immersed into $$\mathbb {R}^{d+2}$$, and which moves along the binormal direction with a speed proportional to its mean curvature. In this article, we prove small data global regularity in low-regularity Sobolev spaces for the skew mean curvature flow in dimensions $$d\geq 4$$. This extends the local well-posedness result in [7].
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Multiscale regression on unknown manifolds
We consider the regression problem of estimating functions on $$ \mathbb{R}^D $$ but supported on a $ d $-dimensional manifold $$ \mathcal{M} ~~\subset \mathbb{R}^D $$ with $$ d \ll D $$. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on $$ \mathcal{M} $$ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $$ \mathbb{R}^D $$. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.
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- PAR ID:
- 10296149
- Date Published:
- Journal Name:
- Mathematics in Engineering
- Volume:
- 4
- Issue:
- 4
- ISSN:
- 2640-3501
- Page Range / eLocation ID:
- 1 to 25
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract The skew mean curvature flow is an evolution equation forddimensional manifolds embedded in$${{\mathbb {R}}}^{d+2}$$ (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension$$d\ge 4$$ .more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3934/mine.2022028
