About Me

Hello and welcome to my website! I'm currently a postdoctoral researcher at KTH Royal Institute of Technology as part of the Differential Geometry & General Relativity group. My mentor is Klaus Kröncke. I received my PhD from Queen Mary University of London under the supervision of Reto Buzano and Huy Nguyen. Prior to that, I obtained my master's degree (MASt) from the the University of Warwick and my bachelor's degree (BSc in Mathematics and Computer Science) from King's College London. My research interests include geometric flows, boundary value problems, and special geometric objects such as Einstein manifolds and minimal surfaces.



Contact Info and Research Links

Email: yudowitz@kth.se

Address: KTH Royal Institute of Technology, Lindstedtsvägen 25, 114 28 Stockholm, Sweden

CV, Arχiv Profile, ResearchGate Profile

Research

Overview of My Research

My current research largely focuses on Ricci flow, ∂t g(t) = -2Ric(g(t)), a geometric heat flow originally introduced to uniformize manifolds and then classify them using a surgery procedure. Due to the non-linearity of the equation, one expects to encounter finite time singularities. That is, when the curvature tensor becomes unbounded in some region as t → T. This is a common sort of behavior in non-linear PDEs and some standard problems are to understand the singularity formation by taking “blow-up limits” (zooming in on the singular region) and then classify the possible limits, as well as figuring out how to continue the flow past the singular time.


Gaussian Bounds of Heat Kernels for Evolving Manifolds

If a singularity forms at a “Type I rate”, then it has been shown that blow-ups in this case yield non-flat gradient shrinking Ricci solitons (manifolds which shrink monotonically under Ricci flow). The proofs of these results vitally depend on good control of the exponential weight in Perelman's W-entropy, a monotone quantity which is constant on gradient shrinkers. Since this exponential weight satisfies an adjoint heat equation, control of it comes in the form of “Gaussian bounds” for the heat kernel. Such bounds have been derived under a variety of conditions, using different methods in each case. In a paper with Reto Buzano, I tried to find a more unified approach to proving such Gaussian bounds, which resulted in estimates depending on the evolution of the distance function under Ricci flow, rather than directly on the curvature of the evolving manifold. I also showed these results hold for manifolds evolving under a wider class of flows, ∂t g(t) = -2Sc(g(t)), provided the symmetric 2-tensor Sc satisfies an inequality involving an adjoint heat equation, Bianchi type identity, and comparison to the Ricci tensor.

Bubble Tree Convergence of Shrinking Ricci Solitons

Currently, a full classification of shrinking solitons is unknown. In low dimensions (n=2,3) we know that they are either (quotients of) spheres or cylinders, or Euclidean space. More recently, smooth 4-dimensional Kähler shrinkers have been classified. However, recent compactness theorems indicate blow-ups in dimensions n ≥ 4 can yield shrinkers that are singular themselves which poses another issue to classification. In a paper with Reto Buzano, I investigated this singularity formation, provided the singular set consists of isolated cone points. This involved applying bubble tree analysis to the space of shrinkers with locally bounded energy by performing successive blow-ups around each singular point. This allowed for the proof of an energy identity which, through the Chern-Gauss-Bonnet theorem, shows that any topology lost due to the singularity formation can be recovered by blowing up around the singular points. The bubble tree analysis also yields a local diffeomorphism finiteness theorem, which acts as a qualitative classification theorem. I have also studied how the formation of orbifold points influences the spectrum of the operator associated to the stability of Ricci shrinkers, in particular that it is lower and upper semi-continuous in an appropriate sense. The techniques used to prove this also led (under a technical condition) to showing an asymptotically conical shrinker is unstable if its asymptotic cone is unstable and an inequality quantifying this.

Stability of Manifolds Under Ricci Flow

A different way to view Ricci flow is a dynamical system acting on the space of Riemannian metrics. This then leads to the question of dynamical stability: do perturbations of a fixed point flow back to a fixed point? For Ricci flow, the fixed points (modulo scaling and diffeomorphisms) are Ricci solitons, which can be viewed as generalizations of Einstein manifolds. This is well understood in the compact case, in large part due to the existence of a good variational theory for the so called shrinker/expander entropies and the lambda functional. This then leads to Łojasiewicz-Simon inequalities, which are powerful tools in proving dynamical stability. This is much harder in the non-compact case due to a lack of well-defined and finite functionals. However, some modifications have been shown to be reasonably well-behaved for certain Ricci-flat ALE manifolds and asymptotically hyperbolic manifolds. In the latter setting, Klaus Kröncke and I have proved, via a Łojasiewicz-Simon inequality, dynamical stability of Poincaré-Einstien manifolds (asymptotically hyperbolic Einstein manifolds). This is determined by whether or not the original metric is a local maximum of the entropy. This hinged on proving good Fredholm properties of the stability operator in this setting, which could potentially break down if the manifold in non-compact. Klaus and I also proved stability is equivalent to local positive mass and volume comparison properites.



Publications

Here is a list of my preprints and published papers. You can click on a paper to see the abstract, links to access the arχiv and journal versions, and occasionally some extra comments.


We prove lower and upper semi-continuity of the Morse index for sequences of gradient Ricci shrinkers which bubble tree converge in the sense of past work by the author and Buzano. Our proofs rely on adapting recent arguments of Workman which were used to study certain sequences of CMC hypersurfaces and were in turn adapted from work of Da Lio-Gianocca-Rivière. Moreover, we are able to refine Workman's methods by using techniques related to polynomially weighted Sobolev spaces. This all also requires us to extend the analysis to handle when the shrinkers we study are non-compact, which we can do due to the availability of a suitable notion of finite weighted volume. Finally, we identify a technical condition which ensures the Morse index of an asymptotically conical shrinker is bounded below by the f-index of its asymptotic cone.

[Arχiv]


We prove dynamical stability and instability theorems for Poincaré-Einstein metrics under the Ricci flow. Our key tool is a variant of the expander entropy for asymptotically hyperbolic manifolds, which Dahl, McCormick and the first author established in a recent article. It allows us to characterize stability and instability in terms of a local positive mass theorem and in terms of volume comparison for nearby metrics.

[Arχiv]


We prove bubble-tree convergence of sequences of gradient Ricci shrinkers with uniformly bounded entropy and uniform local energy bounds, refining the compactness theory of Haslhofer-Müller. In particular, we show that no energy concentrates in neck regions, a result which implies a local energy identity for the sequence. Direct consequences of these results are an identity for the Euler characteristic and a local diffeomorphism finiteness theorem.

[Arχiv, Journal]


In this article, we prove a general and rather flexible upper bound for the heat kernel of a weighted heat operator on a closed manifold evolving by an intrinsic geometric flow. The proof is based on logarithmic Sobolev inequalities and ultracontractivity estimates for the weighted operator along the flow, a method which was previously used by Davies in the case of a non-evolving manifold. This result directly implies Gaussian-type upper bounds for the heat kernel under certain bounds on the evolving distance function; in particular we find new proofs of Gaussian heat kernel bounds on manifolds evolving by Ricci flow with bounded curvature or positive Ricci curvature. We also obtain similar heat kernel bounds for a class of other geometric flows.

[Arχiv, Journal]



Talks, Teaching, and Other Activities

Talks Given

  • Junior Meeting Einstein Geometry and Special Holonomy: “Dynamical Stability and Instability of Poincaré-Einstein Manifolds” (July 25, 2024).
  • KTH Differential Geometry and General Relativity Seminar: “Perelman Functionals for a Class of Intrinsic Geometric Flows” (February 15, 2024).
  • University of Copenhagen Geometry Seminar: “Dynamical Stability and Instability of Poincaré-Einstein Manifolds” (January 24, 2024).
  • KTH Differential Geometry and General Relativity Seminar: “Semi-Continuity of the Morse Index for Ricci Shrinkers” (October 19, 2023).
  • The Crazy World of Arthur L. Besse: A Workshop on Einstein Manifolds: “Bubble Tree Convergence of Shrinking Ricci Solitons” (October 5, 2023).
  • Workshop on Einstein Spaces and Special Geometry, Institut Mittag-Leffler: “Bubble Tree Convergence of Shrinking Ricci Solitons” (July 12, 2023).
  • Ghent Methusalem Junior Seminar: “Bubble Tree Convergence of Shrinking Ricci Solitons” (May 10, 2023).
  • KTH Differential Geometry and General Relativity Seminar: “Bubble Tree Convergence of Gradient Ricci Shrinking Solitons” (Jan. 19, 2023).
  • Brunel University Math and Statistics Colloquium: “Ricci Flow, the Poincaré Conjecture, and Bubbles” (Nov. 16, 2022).
  • KIT Geometric Analysis Seminar: “Bubble Tree Convergence of Gradient Ricci Shrinking Solitons” (Oct. 5, 2022).
  • 9th Heidelberg Laureate Forum: “Bubble Tree Convergence of Gradient Ricci Shrinking Solitons” (Sept. 19, 2022).
  • KCL/UCL Junior Geometry Seminar: “Bubble Tree Convergence of Gradient Ricci Shrinking Solitons” (Jan. 27, 2022).
  • Queen Mary Internal Postgraduate Seminar (QuIPS): “Ricci Flow and the Poincaré Conjecture” (Nov. 2, 2021).

Teaching

KTH Royal Institute of Technology:
  • Degree Project in Mathematical Statistics (as academic supervisor), Spring Semester (2023/2024)
  • Calculus in Several Variables, Fall Semester (2023/2024, 2024/2025)
Teaching Associate, Queen Mary University of London:
  • Probability and Statistics I, Fall Semester (2022/2023)
  • Calculus II, Spring Semester (2021/2022)
  • Actuarial Mathematics I, Fall Semester (2021/2022, 2022/2023)
  • Vectors and Matrices, Spring Semester (2019/2020)