arXiv subjects

Shunzi Guo

Publications and source records attributed to Shunzi Guo.

4 recordsLinked to original sources

arXiv2026

Length-constrained curve diffusion flow for open curves with endpoints on two intersecting lines

We study the curve diffusion flow for open planar curves whose endpoints are constrained to lie on two fixed straight lines that intersect at an angle $\theta (\in(0,\pi)) $. For every such angle, we prove that under suitable initial conditions the flow exists globally in time. Moreover, we show that the evolving curve converges - exponentially and in the smooth topology - to the circular arc of a sector whose central angle is exactly $\theta$ and whose arc length equals that of the initial curve. This result reveals how a length-preserving fourth-order geometric flow can straighten out a curve's shape while respecting boundary constraints, ultimately driving it toward a unique equilibrium: the circular arc spanning the prescribed angle. This provides a complete description of the long-time behaviour of this fourth-order geometric flow with mixed boundary conditions.

math.DG↗

arXiv2016

Contracting convex hypersurfaces by functions of the mean curvature

This paper concerns the evolution of a closed convex hypersurface in ${\mathbb{R}}^{n+1}$, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This result covers and generalises the corresponding result of Schulze \cite{Sch05} for the positive power mean curvature flow to a much larger possible class of flows by the functions depending only on the mean curvature.

math.DG↗

arXiv2016

Mixed volume preserving flow by powers of homogeneous curvature functions of degree one

This paper concerns the evolution of a closed hypersurface of dimension $n(\geq 2)$ in the Euclidean space ${\mathbb{R}}^{n+1}$ under a mixed volume preserving flow. The speed equals a power $β(\geq 1)$ of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, including the case of powers of the mean curvature and of the Gauss curvature. The main result is that if the initial hypersurface satisfies a suitable pinching condition, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces converge exponentially to a round sphere, enclosing the same mixed volume as the initial hypersurface. This result covers and generalises the previous results for convex hypersurfaces in the Euclidean space by McCoy \cite{McC05} and Cabezas-Rivas and Sinestrari \cite{CS10} to more general curvature flows for convex hypersurfaces with similar curvature pinching condition.

math.DG↗

arXiv2013

Volume-Preserving flow by powers of the mth mean curvature in the hyperbolic space

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_κ^{n+1}$ of constant sectional curvature $κ$ evolving in direction of its normal vector, where the speed is given by a power $β(\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the $\mbox{Gauß}$ curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $β$ and $κ$, then under the flow this is maintained, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces exponentially converge to a geodesic sphere of ${\mathbb{H}}_κ^{n+1}$, enclosing the same volume as the initial hypersurface.

math.DG↗