T. Bourni

activity

20102026

most citedA collapsing ancient solution of mean curvature flow in

R^{3}

15 citations · 29 across the 28 of their papers we have counts for

collaborators

34 papers

math.DG2026

Compact Mean Curvature Flows from Non-Compact Initial Data

Theodora Bourni, Nathan Burns, Marcus Flook

We construct compact, embedded solutions to the mean curvature flow in $R^{n + 1}$ that arise from various non-compact initial data and exist for a finite time.

math.DG2026

Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

Theodora Bourni, Nathan Burns, Mat Langford

We establish a sharp rate of convergence for a free-boundary curve shortening flow in a convex domain in $R^{2}$ which converges in finite time to a round half-point.

math.DG2026

Ancient solutions to free boundary mean curvature flow

Theodora Bourni, Giada Franz

We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minima…

math.DG2026

Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

Theodora Bourni, José M. Espinar, Aakash Mishra

We develop an Aleksandrov reflection framework for a large class of expanding curvature flows in hyperbolic space, with inverse mean curvature flow serving as a model case. The met…

cs.SI2024

On the Relation between Graph Ricci Curvature and Community Structure

Sathyanarayanan Rengaswami, Theodora Bourni, Vasileios Maroulas

The connection between curvature and topology is a very well-studied theme in the subject of differential geometry. By suitably defining curvature on networks, the study of this th…

math.DG2024

Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

Theodora Bourni, Nathan Burns, Spencer Catron

We classify convex ancient curve shortening flows with free boundary on general bounded convex domains.

math.DG2023

Convex Ancient Solutions to Anisotropic Curve Shortening Flow

Theodora Bourni, Benjamin Richards

We construct a translating solution to anisotropic curve shortening flow and show that for a given anisotropic factor $g : S^{1} \to R_{+}$ , and a given direction and speed, this…

math.DG2023

Compact curve shortening flow solutions out of non compact curve

Theodora Bourni, Martin Reiris

We construct a slingshot, that is a compact, embedded solution to curve shortening flow that comes out of a non compact curve and exists for a finite time.

math.DG2023

Ancient Ricci flows of bounded girth

Theodora Bourni, Timothy Buttsworth, Ramiro Lafuente +1

For each $n \geq 3$ , we construct a 'pancake-like', $O (2) \times O (n - 1)$ -invariant ancient Ricci flow with positive curvature operator and bounded "girth", and we determine its asympt…

math.DG2022

Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow

Theodora Bourni, Mat Langford, Stephen Lynch

We make rigorous an old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.

math.DG2022★ 1 cited

Classification of convex ancient free boundary mean curvature flows in the ball

Theodora Bourni, Mat Langford

We prove that there exists, in every dimension, a unique (modulo rotations about the origin and time translations) convex ancient mean curvature flow in the ball with free boundary…

math.DG2022

Nonplanar ancient curve shortening flows in $R^{3}$ from grim reapers

Theodora Bourni, Alexander Mramor

In this note we construct new nonplanar ancient (in fact, eternal) solutions to the curve shortening flow in $R^{3}$ , built out of translating grim reapers laying in perpend…

math.DG2021★ 1 cited

Classification of convex ancient free boundary curve shortening flows in the disc

Theodora Bourni, Mat Langford

We classify convex ancient curve shortening flows in the disc with free boundary on the circle.

math.DG2021

Collapsing and noncollapsing in convex ancient mean curvature flow

Theodora Bourni, Mat Langford, Stephen Lynch

We provide several characterisations of collapsing and noncollapsing in convex ancient mean curvature flow, establishing in particular that collapsing occurs if and only if the flo…

math.DG2021

CMC hypersurfaces with bounded Morse index

Theodora Bourni, Ben Sharp, Giuseppe Tinaglia

We develop a bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular we show that c…

math.DG2020★ 3 cited

The atomic structure of ancient grain boundaries

Theodora Bourni, Mat Langford, Giuseppe Tinaglia

Democritus and the early atomists held that "the material cause of all things that exist is the coming together of atoms and void. Atoms are eternal and have many different shapes,…

math.DG2020

The vanishing of the fundamental gap of convex domains in $H^{n}$

Theodora Bourni, Julie Clutterbuck, Xuan Hien Nguyen +3

For the Laplace operator with Dirichlet boundary conditions on convex domains in $H^{n}$ , $n \geq 2$ , we prove that the product of the fundamental gap with the square of the d…

math.DG2020

Ancient solutions for flow by powers of the curvature in $R^{2}$

Theodora Bourni, Julie Clutterbuck, Xuan Hien Nguyen +3

We construct a new compact convex embedded ancient solution of the $κ^{α}$ flow in $R^{2}$ , $α \in (\frac{1}{2}, 1)$ that lies between two parallel lines. Using this solution we class…

math.DG2019★ 1 cited

Differential Harnack inequalities via Concavity of the arrival time

Theodora Bourni, Mat Langford

We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these…

math.DG2019

Explicit fundamental gap estimates for some convex domains in $H^{2}$

Theodora Bourni, Julie Clutterbuck, Xuan Hien Nguyen +3

Motivated by an example of Shih, we compute the fundamental gap of a family of convex domains in the hyperbolic plane $H^{2}$ , showing that for some of them $λ_2 - λ_1 < \fra…

math.DG2019

On the construction of closed nonconvex nonsoliton ancient mean curvature flows

Theodora Bourni, Mathew Langford, Alexander Mramor

We construct closed, embedded, ancient mean curvature flows in each dimension $n \geq 2$ with the topology of $S^{1} \times S^{n - 1}$ . These examples are not mean convex and not soliton…

math.DG2019

Convex ancient solutions to mean curvature flow

Theodora Bourni, Mat Langford, Giuseppe Tinaglia

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,…

math.DG2019

Convex ancient solutions to curve shortening flow

Theodora Bourni, Mat Langford, Giuseppe Tinaglia

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…

math.DG2018★ 2 cited

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

Theodora Bourni, Mat Langford, Giuseppe Tinaglia

We prove, in all dimensions $n \geq 2$ , that there exists a convex translator lying in a slab of width $π sec θ$ in $R^{n + 1}$ (and in no smaller slab) if and only if $θ\in[0…

Compact Mean Curvature Flows from Non-Compact Initial Data

Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

Ancient solutions to free boundary mean curvature flow

Aleksandrov reflection for Geometric Flows in Hyperbolic Spaces

On the Relation between Graph Ricci Curvature and Community Structure

Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

Convex Ancient Solutions to Anisotropic Curve Shortening Flow

Compact curve shortening flow solutions out of non compact curve

Ancient Ricci flows of bounded girth

Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow

Classification of convex ancient free boundary mean curvature flows in the ball

Nonplanar ancient curve shortening flows in R3 from grim reapers

Classification of convex ancient free boundary curve shortening flows in the disc

Collapsing and noncollapsing in convex ancient mean curvature flow

CMC hypersurfaces with bounded Morse index

The atomic structure of ancient grain boundaries

The vanishing of the fundamental gap of convex domains in Hn

Ancient solutions for flow by powers of the curvature in R2

Differential Harnack inequalities via Concavity of the arrival time

Explicit fundamental gap estimates for some convex domains in H2

On the construction of closed nonconvex nonsoliton ancient mean curvature flows

Convex ancient solutions to mean curvature flow

Convex ancient solutions to curve shortening flow

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

Nonplanar ancient curve shortening flows in $R^{3}$ from grim reapers

The vanishing of the fundamental gap of convex domains in $H^{n}$

Ancient solutions for flow by powers of the curvature in $R^{2}$

Explicit fundamental gap estimates for some convex domains in $H^{2}$