Reto Buzano (Mueller), School of Mathematical Sciences, Queen Mary University of London

What is my Research about?


Geometric Analysis: This is an area of mathematics where geometric objects are studied with analytical methods, often involving partial differential equations. As these equations are generally nonlinear, it is typical for singularities to occur in solutions, thus current efforts aim to transform the theory from one where solutions are required to be smooth to one where singularities play a central role. In fact, the presence of singularities often has deep and insightful geometric reasons. My research focusses on various different projects involving singularities in Geometric Analysis, my main emphasis being on geometric heat flows and critical points of geometric functionals.

Heat flow methods have become an important and exciting tool in mathematics, the motivation being to evolve rough initial data towards nice objects, e.g. manifolds with constant curvature, harmonic maps or minimal surfaces. Such flows, and in particular the Ricci Flow, have proved spectacularly successful with Perelman's resolutions of both the Poincaré and Thurston's Geometrisation Conjectures and the Differentiable Sphere Theorem obtained by Brendle-Schoen. My work is mainly on the Ricci Flow, the Mean Curvature Flow, and the Harmonic Ricci Flow (a coupled flow of Ricci Flow and Harmonic Map flow which I introduced). An overview of my past results on geometric flows is given below.

Another category of problems I am interested in comes from the study of geometric functionals and their critical points. Recently, I have mainly been interested in minimal surfaces and conformal geometry. There has been substantial progress in these areas over the past years, especially in two dimensions, the most prominent example of this progress probably being the resolution of the Willmore Conjecture by Marques-Neves. In higher dimensions, where my main interest lies, often even very basic questions are still open. Below, I will describe my work on the Paneitz Q-curvature, leading in particular to new Chern-Gauss-Bonnet formulas for singular metrics, as well as my projects on minimal hypersurfaces with bounded Morse index.

All references below refer to the articles and preprints on my publications page.



Geometric Flows


The Ricci Flow: The Ricci Flow can be seen as the natural heat equation for a Riemannian metric: It shrinks the metric where its (Ricci) curvature is positive and expands it where its curvature is negative. It is well known that this flow typically develops singularities in finite time and fully understanding them is one of the main obstacles in the Ricci Flow theory. In joint work with Joerg Enders and Peter Topping [3], we proved Hamilton's Conjecture that fast-forming (Type I) singularities are always modelled on self-similar solutions in any dimension. As a corollary, we proved that the scalar curvature blows up at a Type I singularity, a result which was also previously conjectured. To obtain more information about the geometric shape of the singularity model, in a joint work with Carlo Mantegazza [5], we studied Perelman's W-entropy functional at a Type I singularity and proved that no entropy is lost in the blow-up process, a result which restricts the possible singularity models one might obtain in a Ricci Flow. In joint work with Robert Haslhofer [4], we showed that the space of singularity models obtained by a blow-up procedure as above is compact. In fact, we obtained a precompactness result (allowing orbifold-singularities) even in the general case without Type I curvature bounds. Later, we further improved our result in the four-dimensional case, getting rid of all the previous assumptions except the necessary one of a lower bound on Perelman’s entropy, see [7]. In a further project with Robert Haslhofer, we also studied the dynamical stability and instability of Ricci-flat manifolds under the Ricci Flow. Ricci-flat manifolds are the stationary points of the Ricci Flow and they are also the critical points of Perelman's energy functional λ. In our article [6], we improved Haslhofer's previous stability and instability results, getting rid of his integrability assumption. Finally, in a very recent preprint with Di Matteo [16], we introduced a local singularity analysis and localised a variety of the above results as well as other important results for the Ricci flow, such as for example Sesum's result that the Ricci curvature blows up at a finite time singularity.

The Harmonic Ricci Flow: Ricci Flow type systems of equations arise naturally from physical problems, e.g. as Renormalisation Group flows in Quantum Field Theory or from questions about Einstein vacuum equations in General Relativity. Such flow systems have recently become more popular and it is natural to ask how Perelman's influential work on Ricci Flow can be adapted to them. The Harmonic Ricci Flow is an instance of such a flow system, a coupling of Ricci Flow with Harmonic Map Flow which I introduced in [2]. I first extended Perelman's reduced volume monotonicity result to this coupled flow [1], leading also to new monotonicity formulas for List's extended Ricci flow system as a special case and for the Mean Curvature Flow of spacelike hypersurfaces in Lorentzian manifolds. In [2], I then showed that a large portion of the Ricci Flow techniques (energy, entropy, non-collapsing, no breather theorem, etc.) carry over to the coupled flow system, but moreover this system also behaves in a less singular way than the Ricci Flow or the Harmonic Map Flow considered alone in some situations. In particular, for large values of some coupling constant, I ruled out singularities induced by an energy concentration of the map, thus long-time existence is obtained once one can exclude curvature singularities. Together with Melanie Rupflin [9], we then proved that the curvature must indeed stay bounded along the Harmonic Ricci Flow on surfaces. With Yudowitz, we then obtained Gaussian bounds [17] for the heat kernel on manifolds evolving by a class of geometric flows including the Ricci flow and the Harmonic Ricci flow.

The Mean Curvature Flow: In joint work with Robert Haslhofer and Or Hershkovits, we investigated the moduli space of embedded n-spheres in Euclidean (n+1)-space. By Smale's Theorem and Hatcher's proof of Smale's Conjecture, one knows that this moduli space is contractible in dimensions one and two, but it is also known that this is no longer the case in dimensions four or higher. The three-dimensional case is related to various deep open poblems in topology, in particular the Schoenflies Conjecture and the Smooth 4D Poincaré Conjecture. We conjecture that if one restricts to two-convex spheres, the space should always be contractible in any dimension and in our article [11] we verified the π_0 part of this Smale Type Conjecture, namely, we showed that this moduli space is path-connected. Our method is to construct a modified Mean Curvature Flow with surgery using a variant of a two-convex connected sum construction where we carefully glue in tiny cylindrical strings in regions where surgery occurred in order to re-connect the hypersurface and thus produce a flow of embedded spheres. This connected sum construction can be seen as an extrinsic analog of the famous Gromov–Lawson construction and our main result can be seen as an extrinsic variant of Marques' path-connectedness theorem for positive scalar curvature metrics on three-manifolds. In a second article [13], we have then extended our results to study embedded knotted tori. The main theorem here states that two embedded knotted two-convex tori are in the same connected component of the moduli space if and only if they represent the same knot class.



Geometric Functionals and their Critical Points


Singular Conformal Geometry: Many interesting objects in geometry arise as critical points of some geometric functional in the sense of calculus of variations, examples include Einstein manifolds, harmonic maps, Willmore surfaces, etc. Often the functionals considered have natural geometric invariances, for example invariance under diffeomorphisms or under conformal transformations. An easy but fundamental example for such a functional is the total Gauss curvature in dimension two: The Gauss-Bonnet theorem, one of the most fundamental results in differential geometry, gives a link between the geometry of a surface (given by its total Gauss curvature) and its topology (given by its Euler characteristic). In particular, it shows that there are topological obstructions to the existence of certain metrics, for example no two-dimensional torus carries a metric of positive Gauss curvature. A generalisation of the Gauss-Bonnet theorem to higher-dimensional compact Riemannian manifolds was discovered by Chern and has been known for over fifty years. However, little is known about the corresponding formula for non-compact or singular Riemannian manifolds. In joint work with Huy Nguyen [8], we proved a new four-dimensional Chern-Gauss-Bonnet formula for metrics with finitely many conformally flat ends and singular points, under very weak (and necessary) curvature restrictions. This is the first such result in a dimensions higher than two which allows the underlying manifold to have isolated branch points or conical singularities. The formula which we obtained includes a precise characterisation of the error terms, expressing them as isoperimetric deficits near the singular points, infinitesimally measuring the deviation from flat Euclidean space. In a more recent project [12], we generalised our results to dimensions higher than four in the locally conformally flat case.

Minimal Hypersurfaces: An even simpler geometric functional than the total curvature discussed above is the area functional for a submanifold. Its critical points are minimal surfaces (and its gradient flow is the Mean Curvature Flow). One can find many closed, smooth, embedded minimal hypersurfaces in a closed Riemannian manifold N via min-max constructions, using the theory of Almgren-Pitts or Allan-Cahn equations. Due to the min-max construction, these minimal hypersurfaces have bounded Morse index and bounded area and it thus seems natural to study the properties of minimal hypersurfaces satisfying these two conditions. While it was known for over twenty years that the Morse index is bounded linearly from above in terms of area and total curvature, results for lower bounds were unknown until recently. In joint work with Ben Sharp [10], we gave qualitative lower bounds on index and area in terms of total curvature for embedded minimal n-dimensional hypersurfaces in a closed Riemannian (n+1)-manifold in dimensions 2≤n≤6. Our proof uses a bubbling argument, and as a consequence we also obtain a concentration-compactness result for the space of such minimal hypersurfaces and in particular an energy identity for the total curvature. Together with Lucas Ambrozio, Alessandro Carlotto, and Ben Sharp [14], we used this energy identity to obtain new smooth multiplicity one compactness theorems for minimal surfaces, in particular generalising classical results of Choi-Schoen from ambient spaces N with positive Ricci curvature, to ambient spaces with positive scalar curvature under an additional bound on either the index or the area. We also confirmed the folklore conjecture that under smooth graphical convergence away from finitely many points (and with finite multiplicity m≥1) the genus can only drop. Finally, we also obtained a sharp estimate on the multiplicity of the convergence in terms of the number of ends of the bubbles. In a second article [15], we then generalised all of these results to free boundary minimal hypersurfaces, in particular analysing the formation of "half-bubbles", which are free boundary minimal hypersurfaces living in a half-space of Euclidean (n+1)-space.



Geometric Analysis at QMUL and UniTo


Current Group: I am leading the Geometric Analysis Group at Queen Mary University together with Huy Nguyen. Besides us, this group currently consists of two postdoctoral researchers (Mario Schulz and Shengwen Wang) as well as four PhD students (Florian Litzinger and Gianmichele Di Matteo, who have started in September 2017, as well as Quintin Luong and Louis Yudowitz who have started in September 2019). We have our own weekly Research Seminar and Reading Seminar at QMUL, but we also actively participate in the Geometric Analysis Reading Seminar at UCL. Moreover, I am an organiser of the Brussels-London Geometry Seminar. Further information about our research group can be found on the dedicated group webpage at geometricanalysis.london.

At the University of Torino, where I am a member of the Geometry group led by Anna Fino, I currently do not have postdocs or PhD students yet. If you are interesed in working with me there, please contact me.