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Weingarten Flows in Riemannian Manifolds
Differential Geometry · Submitted 19 May 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that the avoidance principle holds for Weingarten flows defined by odd Weingarten functions, and also that such flows are embedding preserving.
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The abstract, sentence by sentence
Given orientable Riemannian manifolds $M^n$ and $\bar M^{n+1},$ we study flows $F_t:M^n\rightarrow\bar M^{n+1},$ called Weingarten flows,in which the hypersurfaces $F_t(M)$ evolve in the direction of their normal vectors with speed given by a function $W$ of their principal curvatures,called a Weingarten function, which is homogeneous, monotonic increasing with respect to any of its variables, and positive on the positive cone. We obtain existence results for flows with isoparametric initial data, in which the hypersurfaces $F_t:M^n\rightarrow\bar M^{n+1}$ are all parallel, and $\bar M^{n+1}$ is either a simply connected space form or a rank-one symmetric space of noncompact type. We prove that the avoidance principle holds for Weingarten flows defined by odd Weingarten functions, and also that such flows are embedding preserving.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Given orientable Riemannian manifolds $M^n$ and $\bar M^{n+1},$ we study flows $F_t:M^n\rightarrow\bar M^{n+1},$ called Weingarten flows,in which the hypersurfaces $F_t(M)$ evolve in the direction of their normal vectors with speed given by a function $W$ of their principal curvatures,called a Weingarten function, which is homogeneous, monotonic increasing with respect to any of its variables, and positive on the positive cone. | contribution 0.64 | 0.42 | 0.07 | asserted 0.96 |
| s1 | We obtain existence results for flows with isoparametric initial data, in which the hypersurfaces $F_t:M^n\rightarrow\bar M^{n+1}$ are all parallel, and $\bar M^{n+1}$ is either a simply connected space form or a rank-one symmetric space of noncompact type. | result 0.97 | 0.95 | 0.25 | asserted 1.00 |
| s2 | We prove that the avoidance principle holds for Weingarten flows defined by odd Weingarten functions, and also that such flows are embedding preserving. | result 0.99 | 0.95 | 0.22 | asserted 1.00 |
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Original abstract
Given orientable Riemannian manifolds $M^n$ and $\bar M^{n+1},$ we study flows $F_t:M^n\rightarrow\bar M^{n+1},$ called Weingarten flows,in which the hypersurfaces $F_t(M)$ evolve in the direction of their normal vectors with speed given by a function $W$ of their principal curvatures,called a Weingarten function, which is homogeneous, monotonic increasing with respect to any of its variables, and positive on the positive cone. We obtain existence results for flows with isoparametric initial data, in which the hypersurfaces $F_t:M^n\rightarrow\bar M^{n+1}$ are all parallel, and $\bar M^{n+1}$ is either a simply connected space form or a rank-one symmetric space of noncompact type. We prove that the avoidance principle holds for Weingarten flows defined by odd Weingarten functions, and also that such flows are embedding preserving.
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1 arXiv paper cite this in 6 sentences
Papers citing it · 4 sentences without a confident label
Builds on it 1 paper
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Our result provides an extension to general ambient spaces of [ref] , and an extension of [ref] to general isoparametric hypersurfaces, when the MCF is considered.
Isoparametric hypersurfaces of Riemannian manifolds as initial data for the mean curvature flow · Introduction · 1.00
Cites it as background 1 paper
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In this case, following [ref] , isoparametric hypersurfaces are defined in [ref] as those whose parallels have constant principal curvatures, thus including the case in which the ambient spaces are space forms.
Isoparametric hypersurfaces of Riemannian manifolds as initial data for the mean curvature flow · Introduction · 0.86
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