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Theodora Bourni

Publications and source records attributed to Theodora Bourni.

34 records · Page 2Linked to original sources
arXiv2020

Ancient solutions for flow by powers of the curvature in $\mathbb R^2$

We construct a new compact convex embedded ancient solution of the $κ^α$ flow in $\mathbb R^2$, $α\in(\frac12,1)$ that lies between two parallel lines. Using this solution we classify all convex ancient solutions of the $κ^α$ flow in $\mathbb R^2$, for $α\in(\frac23,1)$. Moreover, we show that any non-compact convex embedded ancient solution of the $κ^α$ flow in $\mathbb R^2$, $α\in(\frac12,1)$ must be a translating solution.

math.DG↗
arXiv2019

Explicit fundamental gap estimates for some convex domains in $\mathbb H^2$

Motivated by an example of Shih, we compute the fundamental gap of a family of convex domains in the hyperbolic plane $\mathbb H^2$, showing that for some of them $λ_2 - λ_1 < \frac{3π^2}{D^2}$, where $D$ is the diameter of the domain and $λ_1$, $λ_2$ are the first and second Dirichlet eigenvalues of the Laplace operator on the domain. The result contrasts with what is known in $\mathbb R^n $ or $\mathbb S^n$, where $λ_2 - λ_1 \geq \frac{3 π^2}{D^2}$ for convex domains. We also show that the fundamental gap of the example in Shih's article is still greater than $\tfrac 32 \frac{π^2}{D^2}$, even though the first eigenfunction of the Laplace operator is not log-concave.

math.DG↗
arXiv2019

On the construction of closed nonconvex nonsoliton ancient mean curvature flows

We construct closed, embedded, ancient mean curvature flows in each dimension $n\ge 2$ with the topology of $S^1 \times S^{n-1}$. These examples are not mean convex and not solitons. They are constructed by analyzing perturbations of the self-shrinking doughnuts constructed by Drugan and Nguyen (or, alternatively, Angenent's self shrinking torus when $n =2$)

math.DG↗
arXiv2019

Convex ancient solutions to mean curvature flow

X.-J. Wang proved a series of remarkable results on the structure of convex ancient solutions to mean curvature flow. Some of his results do not appear to be widely known, however, possibly due to the technical nature of his arguments and his exploitation of methods which are not widely used in mean curvature flow. In this expository article, we present Wang's structure theory and some of its consequences. We shall simplify some of Wang's analysis by making use of the monotonicity formula and the differential Harnack inequality, and obtain an important additional structure result by exploiting the latter. We conclude by showing that various rigidity results for convex ancient solutions and convex translators follow quite directly from the structure theory, including the new result of Corollary 8.3}. We recently provided a complete classification of convex ancient solutions to curve shortening flow by exploiting similar arguments.

math.DG↗
arXiv2019

Convex ancient solutions to curve shortening flow

We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Sesum and X.-J. Wang

math.DG↗
arXiv2018

On the existence of translating solutions of mean curvature flow in slab regions

We prove, in all dimensions $n\geq 2$, that there exists a convex translator lying in a slab of width $π\secθ$ in $\mathbb{R}^{n+1}$ (and in no smaller slab) if and only if $θ\in[0,\fracπ{2}]$. We also obtain convexity and regularity results for translators which admit appropriate symmetries and study the asymptotics and reflection symmetry of translators lying in slab regions.

math.DG↗
arXiv2017

A collapsing ancient solution of mean curvature flow in $\mathbb{R}^3$

We construct a compact, convex ancient solution of mean curvature flow in $\mathbb R^{n+1}$ with $O(1)\times O(n)$ symmetry that lies in a slab of width $π$. We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, $O(n)$-invariant ancient solution that lies in a slab of width $π$ and in no smaller slab.

math.DG↗
arXiv2017

Null mean curvature flow and outermost MOTS

We study the evolution of hypersurfaces in spacetime initial data sets by their null mean curvature. A theory of weak solutions is developed using the level-set approach. Starting from an arbitrary mean convex, outer untapped hypersurface $\partialΩ_0$, we show that there exists a weak solution to the null mean curvature flow, given as a limit of approximate solutions that are defined using the $\varepsilon$-regularization method. We show that the approximate solutions blow up on the outermost MOTS and the weak solution converges (as boundaries of finite perimeter sets) to a generalized MOTS.

math.DG↗
arXiv2016

Type-II singularities of two-convex immersed mean curvature flow

We show that any strictly mean convex translator of dimension $n\geq 3$ which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension $n\geq 3$ is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular application, we prove an analogous result for a class of flows of embedded hypersurfaces which includes the flow of two-convex hypersurfaces by the two-harmonic mean curvature.

math.DG↗
arXiv2014

Allard-type boundary regularity for $C^{1,α}$ boundaries

In this paper we show boundary monotonicity formulae for rectifiable varifolds having a $C^{1,α}$ "boundary". In particular, we show that the area ratios of balls centered at this "boundary'' satisfy a nice monotonicity formula, similar to that for interior balls (proved in Allard's paper "On the first variation of a varifold''). This extends the boundary monotonicity formulae of Allard (see "On the first variation of a varifold- boundary behavior''), which require that the boundary is $C^{1,1}$. As a corollary, Allard's boundary regularity results extend to this case and provide a regularity result for rectifiable varifolds with a $C^{1,α}$ ``boundary''.

math.AP↗
arXiv2014

$C^{1,α}$-regularity for surfaces with $H$ in $L^p$

In this paper we prove several results on the geometry of surfaces immersed in $\mathbf R^3$ with small or bounded $L^2$ norm of $|A|$. For instance, we prove that if the $L^2$ norm of $|A|$ and the $L^p$ norm of $H$, $p>2$, are sufficiently small, then such a surface is graphical away from its boundary. We also prove that given an embedded disk with bounded $L^2$ norm of $|A|$, not necessarily small, then such a disk is graphical away from its boundary, provided that the $L^p$ norm of $H$ is sufficiently small, $p>2$. These results are related to previous work of Schoen-Simon and Colding-Minicozzi.

math.DG↗
arXiv2013

An Allard type regularity theorem for varifolds with Hölder continuous generalized normal

We prove that Allard's regularity theorem holds for rectifiable $n$-dimensional varifolds $V$ assuming a weaker condition on the first variation. This, in the special case when $V$ is a smooth manifold translates to the following: If $ω_n^{-1}ρ^{-n}{\rm area}(V\cap B_ρ(x))$ is sufficiently close to $1$ and the unit normal of $V$ satisfies a $C^{0,α}$ estimate, then $V\cap B_{ρ/2}(x)$ is the graph of a $C^{1,α}$ function with estimates. Furthermore, a similar boundary regularity theorem is true.

math.AP↗
arXiv2012

Area minimizing surfaces in mean convex 3-manifolds

In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0, the space of simple closed curves in the boundary of M where all the absolutely area minimizing surfaces they bound in M has genus >=g is open and dense in the space A of nullhomologous simple closed curves in the boundary of M. For showing this we prove a bridge principle for absolutely area minimizing surfaces. Moreover, we show that for any g>=0, there exists a curve in A such that the minimum genus of the absolutely area minimizing surfaces it bounds is exactly g. As an application of these results, we further prove that the simple closed curves in the boundary of M bounding more than one minimal surface in M is an open and dense subset of A. We also show that there are disjoint simple closed curves in the boundary of M bounding minimal surfaces in M which are not disjoint. This allows us to answer a question of Meeks, by showing that for any strictly mean convex 3-manifold M, there exists a simple closed curve Γin the boundary of M which bounds a stable minimal surface which is not embedded.

math.DG↗
arXiv2011

Curvature estimates for surfaces with bounded mean curvature

Estimates for the norm of the second fundamental form, $|A|$, play a crucial role in studying the geometry of surfaces. In fact, when $|A|$ is bounded the surface cannot bend too sharply. In this paper we prove that for an embedded geodesic disk with bounded $L^2$ norm of $|A|$, $|A|$ is bounded at interior points, provided that the $W^{1,p}$ norm of its mean curvature is sufficiently small, $p>2$. In doing this we generalize some renowned estimates on $|A|$ for minimal surfaces.

math.DG↗
arXiv2010

$C^{1,α}$ theory for the prescribed mean curvature equation with Dirichlet data

In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To such a solution, we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data and which inherits a natural minimizing property. Our main result is that its support is a $C^{1,α}$ manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class $C^{1,α}$.

math.DG↗