arXiv subjects

Ben Andrews

Publications and source records attributed to Ben Andrews.

44 records · Page 3Linked to original sources

arXiv2011

Proof of the fundamental gap conjecture

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero potential. More generally, for an arbitrary smooth potential in higher dimensions, our proof gives both a sharp lower bound for the spectral gap and a sharp modulus of concavity for the logarithm of the first eigenfunction, in terms of the diameter of the domain and a modulus of convexity for the potential.

math.SP↗

arXiv2010

Moving surfaces by non-concave curvature functions

A convex surface contracting by a strictly monotone, homogeneous degree one function of curvature remains smooth until it contracts to a point in finite time, and is asymptotically spherical in shape. No assumptions are made on the concavity of the speed as a function of principal curvatures.

math.DG↗

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↗

arXiv2009

Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere

We prove a comparison theorem for the isoperimetric profiles of solutions of the normalized Ricci flow on the two-sphere: If the isoperimetric profile of the initial metric is greater than that of some positively curved axisymmetric metric, then the inequality remains true for the isoperimetric profiles of the evolved metrics. We apply this using the Rosenau solution as the model metric to deduce sharp time-dependent curvature bounds for arbitrary solutions of the normalized Ricci flow on the two-sphere. This gives a simple and direct proof of convergence to a constant curvature metric without use of any blowup or compactness arguments, Harnack estimates, or any classification of behaviour near singularities.

math.DG↗

arXiv2009

Curvature bound for curve shortening flow via distance comparison and a direct proof of Grayson's theorem

A new isoperimetric estimate is proved for embedded closed curves evolving by curve shortening flow, normalized to have total length $2π$. The estimate bounds the length of any chord from below in terms of the arc length between its endpoints and elapsed time. Applying the estimate to short segments we deduce directly that the maximum curvature decays exponentially to 1. This gives a self-contained proof of Grayson's theorem which does not require the monotonicity formula or the classification of singularities.

math.DG↗

arXiv2004

Pinching estimates and motion of hypersurfaces by curvature functions

Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earlier results of B. Chow and the author. The result is obtained via a detailed analysis of gradient terms in the equations satisfied by second derivatives.

math.DG↗

arXiv2004

Fully nonlinear parabolic equations in two space variables

Hölder estimates for second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.

math.AP↗

arXiv2003

Positively curved surfaces in the three-sphere

In this talk I will discuss an example of the use of fully nonlinear parabolic flows to prove geometric results. I will emphasise the fact that there is a wide variety of geometric parabolic equations to choose from, and to get the best results it can be very important to choose the best flow. I will illustrate this in the setting of surfaces in a three-dimensional sphere. There are quite a few relevant results for surfaces in the sphere satisfying various kinds of curvature equations, including totally umbillic surfaces, minimal surfaces and constant mean curvature surfaces, and intrinsically flat surfaces. Parabolic flows can strengthen such results by allowing classes of surfaces satisfying curvature inequalities rather than equalities: This was first done by Huisken, who used mean curvature flow to deform certain classes of surfaces to totally umbillic surfaces. This motivates the question ``What is the optimal result of this kind?'' -- that is, what is the weakest pointwise curvature condition which defines a class of surfaces which retracts to the space of great spheres? The answer to this question can be guessed in view of the examples. To prove it requires a surprising choice of evolution equation, forced by the requirement that the pointwise curvature condition be preserved. I will conclude by mentioning some other geometric situations in which strong results can be proved by choosing the best possible evolution equation.

math.DG↗