arXiv subjects

James McCoy

Publications and source records attributed to James McCoy.

23 records · Page 2Linked to original sources

arXiv2012

Finite time singularities for the locally constrained Willmore flow of surfaces

In this paper we study the steepest descent $L^2$-gradient flow of the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, $λ_1$-weighted surface area, and $λ_2$-weighted enclosed volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our first results are a concentration-compactness alternative and interior estimates for the flow. For initial data with small energy, we prove preservation of embeddedness, and by directly estimating the Euler-Lagrange operator from below in $L^2$ we obtain that the maximal time of existence is finite. Combining this result with the analysis of a suitable blowup allows us to show that for such initial data the flow contracts to a round point in finite time.

math.DG↗

arXiv2012

A classification theorem for Helfrich surfaces

In this paper we study the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, $λ_1$-weighted surface area, and $λ_2$-weighted volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all smooth immersed critical points of the functional with $λ_1\ge0$ and small $L^2$ norm of tracefree curvature. In particular we prove the non-existence of critical points of the functional for which the surface area and enclosed volume are positively weighted.

math.DG↗

arXiv2011

Non-collapsing in fully nonlinear curvature flows

We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a non-collapsing estimate: Precisely, the function which gives the curvature of the largest interior sphere touching the hypersurface at each point is a subsolution of the linearized flow equation if the speed is concave. If the speed is convex then there is an analogous statement for exterior spheres. In particular, if the hypersurface moves with positive speed and the speed is concave in the principal curvatures, then the curvature of the largest touching interior sphere is bounded by a multiple of the speed as long as the solution exists. The proof uses a maximum principle applied to a function of two points on the evolving hypersurface. We illustrate the techniques required for dealing with such functions in a proof of the known containment principle for flows of hypersurfaces.

math.DG↗

arXiv2011

Contracting convex hypersurfaces by curvature

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smooth, and the limiting shape at the final time is spherical. We provide a surprisingly large family of flows for which such results fail, by a variety of mechanisms: Uniformly convex hypersurfaces may become non-convex, and smooth ones may develop curvature singularities; even where this does not occur, non-uniformly convex regions and singular parts in the initial hypersurface may persist, including flat sides, ridges of infinite curvature, or `cylindrical' regions where some of the principal curvatures vanish; such cylindrical regions may persist even if the speed is positive, and in such cases the hypersurface may even collapse to a line segment or higher-dimensional disc rather than to a point. We provide sufficient conditions for these various disasters to occur, and by avoiding these arrive at a class of flows for which arbitrary weakly convex initial hypersurfaces immediately become smooth and uniformly convex and contract to points.

math.AP↗

arXiv2009

Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature

We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero. We apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one. For smooth, strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point, we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time.

math.DG↗