Giuseppe Tinaglia’s research while affiliated with Kingston College and other places

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Publications (54)

On the geometry of the asymptotic boundary of translators in $\mathbb H^2\times \mathbb R
  • Preprint

May 2025

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5 Reads

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Joao Paulo dos Santos

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Giuseppe Tinaglia

In this work, we study complete properly immersed translators in the product space H2×R\mathbb H^2\times\mathbb R, focusing on their asymptotic behavior at infinity. We classify the asymptotic boundary components of these translators under suitable continuity assumptions. Specifically, we prove that if a boundary component lies in the vertical asymptotic boundary, it is of the form {p}×[T,∞)\{p\}\times [T,\infty) or {p}×R\{p\}\times \mathbb R, while if it lies in the horizontal asymptotic boundary, it is a complete geodesic. Our approach is inspired by earlier work on minimal and constant mean curvature surfaces in H2×R\mathbb H^2\times\mathbb R, with a key ingredient being the use of symmetric translators as barriers.

Radius estimates for nearly stable H-hypersurfaces of dimension 2, 3, and 4

November 2024

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3 Reads

Giuseppe Tinaglia

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Alex Zhou

In this paper we study the geometry of complete constant mean curvature (CMC) hypersurfaces immersed in an (n + 1)-dimensional Riemannian manifold N (n = 2, 3 and 4) with sectional curvatures uniformly bounded from below. We generalise radius estimates given by Rosenberg [32] (n = 2) and by Elbert, Nelli and Rosenberg [13] and Cheng [2] (n = 3, 4) to nearly stable CMC hypersurfaces immersed in N. We also prove that certain CMC hypersurfaces effectively embedded in N must be proper.

CMC hypersurfaces with bounded Morse index
  • Article
  • Full-text available

April 2022

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24 Reads

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6 Citations

Theodora Bourni

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Ben Sharp

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Giuseppe Tinaglia

We provide qualitative bounds on the area and topology of separating constant mean curvature (CMC) surfaces of bounded (Morse) index. We also develop a suitable bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular, we show that convergence always occurs with multiplicity one, which implies that the minimal blow-ups (bubbles) are all catenoids.

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A right-handed 3-valued graph
The point P is at distance r from the x3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_3$$\end{document}-axis. Given t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document}, the thick line represents a vertical section of the topological boundary of BD(P,2r+t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^D(P,2r+t)$$\end{document} containing P and the x3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_3$$\end{document}-axis. For s∈(0,r+t]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (0,r+t]$$\end{document}, the intersection BD(P,2r+t)∩{x3=x3(P)±s}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^D(P,2r+t)\cap \{x_3=x_3(P)\pm s\} $$\end{document} consists of an open disk of radius r+t-s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r+t-s$$\end{document} centered at (0,0,x3(P)±s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0,0,x_3(P)\pm s)$$\end{document}. The intersection BD(P,2r+t)∩{x3=x3(P)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B^D(P,2r+t)\cap \{x_3=x_3(P)\}$$\end{document} consists of the open disk centered at P of radius 2r+t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2r+t$$\end{document}. It is easy to see that if Q∉BD(P,2r+t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q\notin B^D(P,2r+t)$$\end{document} then |Q-P|≥22t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|Q-P|\ge \frac{\sqrt{2}}{2} t$$\end{document}
Limit lamination theorem for H-disks

November 2021

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48 Reads

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12 Citations

Inventiones mathematicae

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Giuseppe Tinaglia

In this paper we prove a theorem concerning lamination limits of sequences of compact disks MnM_n embedded in R3\mathbb{R}^3 with constant mean curvature HnH_n, when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi in [8]. We apply this theorem to prove the existence of a chord arc result for compact disks embedded in R3\mathbb{R}^3 with constant mean curvature; this chord arc result generalizes Theorem 0.5 by Colding and Minicozzi in [9] for minimal disks.

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CMC hypersurfaces with bounded Morse index

February 2021

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38 Reads

Theodora Bourni

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Ben Sharp

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Giuseppe Tinaglia

We provide qualitative bounds on the area and topology of separating constant mean curvature (CMC) surfaces of bounded (Morse) index. We also develop a suitable bubble-compactness theory for embedded CMC hypersurfaces with bounded index and area inside closed Riemannian manifolds in low dimensions. In particular we show that convergence always occurs with multiplicity one, which implies that the minimal blow-ups (bubbles) are all catenoids.

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Convex Ancient Solutions to Mean Curvature Flow

October 2020

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4 Reads

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7 Citations

Theodora Bourni

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Mat Langford

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Giuseppe Tinaglia

The 2019 'Australian-German Workshop on Differential Geometry in the Large' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures on manifolds, non-negative curvature and Alexandrov geometry, and topics in differential topology. A joy to the expert and novice alike, this proceedings volume touches on topics as diverse as Ricci and mean curvature flow, geometric invariant theory, Alexandrov spaces, almost formality, prescribed Ricci curvature, and Kähler and Sasaki geometry.

Citations (38)

... • (Angenent-Daskalopoulos-Sesum) The only ancient oval in R 3 is the Z 2 × O(2)-symmetric oval from [Whi03,HH16], which is indeed a 1-oval, and the only 1-oval in R 4 is the Z 2 ×O(3)-symmetric oval from [Whi03,HH16]. • (Choi-Daskalopoulos-Du-Haslhofer-Sesum) Every 2-oval in R 4 is either the unique O(2)×O(2)-symmetric oval from [Whi03,HH16] or a member of the 1-parameter family of Z 2 2 × O(2)-symmetric ovals from [DH]. 2 We mention there are studies on compact ancient collapsed flows, known as pancakes, [BLT21,BLT22]. Formally, they correspond to the case k = n. ...

Reference:

Rigidity of ancient ovals in higher dimensional mean curvature flow
Ancient mean curvature flows out of polytopes
  • Citing Article

  • October 2022

Geometry & Topology

Theodora Bourni

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Mat Langford

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Giuseppe Tinaglia

... The multiplicity analysis in the free boundary case is similar to the closed case in [7, §4]. We follow the original ideas in the proof of [7,Theorem 4.1] but we address a minor mistake found in the proof of [7,Claim 4.5] (see also Remark 5.2). ...

Reference:

Compactness of the space of free boundary CMC surfaces with bounded topology
CMC hypersurfaces with bounded Morse index
Theodora Bourni

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Ben Sharp

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Giuseppe Tinaglia

... Topologically, M is diffeomorphic to S n−1 × S 1 . The mean curvature of M , with respect to the inward-pointing normal, is constant with respect to the S n−1 parameters and is a function only of θ (see [BLT21,§3] [AAG95], which we adjust to our convention of rotating around the y-axis): ...

Reference:

An Intersection Principle for Mean Curvature Flow
Collapsing ancient solutions of mean curvature flow
  • Citing Article

  • October 2021

Journal of Differential Geometry

Theodora Bourni

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Mat Langford

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Giuseppe Tinaglia

... There are many works aimed at understanding these solutions both in the Euclidean case (e.g. [2], [12], [21], [22], [15], among many others) and for more general ambient spaces, see for instance [13,14,32,36,37,29] and the references in [8]. More precisely, a self-similar solution is an evolution that is determined by the flow of a conformal Killing vector field V , i.e. a vector field that satisfies L V g = Cg, for some function C. Due to this additional assumption, (1) reduces to an elliptic equation. ...

Reference:

The mean curvature flow of subgroups on Lie groups of dimension three
Translating Solutions to Mean Curvature Flow
  • Citing Chapter

  • May 2021

Theodora Bourni

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Mat Langford

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Giuseppe Tinaglia

... By the results of Wang [31], convex ancient solutions of the mean curvature flow either sweep the whole space, or are confined in the slab between two parallel hyperplanes. Following [10], we call the former ones (except for the round sphere) ovaloids, and the latter ones pancakes. The existence of ovaloids was first shown by White [32], who sketched an approximation procedure by a sequence of convex solutions of increas-ing eccentricity. ...

Reference:

Non-homothetic convex ancient solutions for flows by high powers of curvature
Convex Ancient Solutions to Mean Curvature Flow
  • Citing Chapter

  • October 2020

Theodora Bourni

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Mat Langford

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Giuseppe Tinaglia

... Namely, one may first deform the helicoid very slightly to a surface with H > 0, and in fact with H uniformly bounded away from 0 and ∞, cf. [17] for instance. The vertical cylinder has H > 0 and one can bend the top flat disc to H > 0. Finally, the corners of S 2 may also be smoothed to H > 0 everywhere. ...

Reference:

Boundary Value Problems for Metrics on 3-manifolds
Multi-valued graphs in embedded constant mean curvature disks
  • Citing Article

  • August 2006

Transactions of the American Mathematical Society

Giuseppe Tinaglia

... Thus, the classifcation of all ancient solutions is a useful tool in the study of the behavior of a flow. In the case of CSF, compact, convex ancient solutions were classified by Daskalopoulos-Hamilton-Sesum [6], and this classification was extended to all convex cuves by Bourni-Langford-Tinaglia [5]. Classification of convex ancient solutions for curves solving a flow based on the curvature raised to certain powers was done by Bourni et. ...

Reference:

Convex ancient solutions to anisotropic curve shortening flow
Convex ancient solutions to curve shortening flow

Calculus of Variations and Partial Differential Equations

Theodora Bourni

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Mat Langford

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Giuseppe Tinaglia

... J. Wang proved in [Wan11] that any convex translator in R n+1 which is not an entire graph lies in a vertical slab of finite width (i.e. is collapsed). Hence, using the grim reaper cylinder as a barrier, the following result can be shown (see e.g [HIMW19], [BLT20]). ...

Reference:

Rigidity and non-existence results for collapsed translators
On the existence of translating solutions of mean curvature flow in slab regions
  • Citing Article

  • June 2020

Analysis and Partial Differential Equations

Theodora Bourni

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Mat Langford

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Giuseppe Tinaglia

... Meeks and Rosenberg generalised this to complete minimal surfaces embedded in R 3 with positive injectivity radius [21], see also [20]. Finally, Meeks and Tinaglia further generalised both these results to constant mean curvature (CMC) surfaces [24], see also [23,25,26,27,28]. ...

Reference:

Radius estimates for nearly stable H-hypersurfaces of dimension 2, 3, and 4
Curvature estimates for constant mean curvature surfaces
  • Citing Article

  • October 2019

Duke Mathematical Journal

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Giuseppe Tinaglia

... By analyzing the asymptotics of this solution, we are able to use Alexandrov reflection principle to show first that any such solution is reflection symmetric with respect to the mid-plane of the slab and second, for α ∈ ( 2 3 , 1], that the solution is unique. Many of the techniques used in the construction as well as in the proof of uniqueness of this solution, have been used in [3,4]. A major difficulty encountered for α < 1 is that the derivative of the enclosed area is no longer constant, which was a crucial ingredient in the previous works. ...

Reference:

Ancient solutions for flow by powers of the curvature in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^2$$\end{document}
A collapsing ancient solution of mean curvature flow in $\mathbb{R}^3
  • Citing Article

  • May 2017

Theodora Bourni

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Mat Langford

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Giuseppe Tinaglia