# UCSB Differential Geometry Seminar

The talks are held every Friday 3–4 PM (Pacific Time) unless otherwise noted. If you are not a member of the UCSB Math Department, but would like to attend one of our talks, then please email me.

# Upcoming talks

## Spring 2023

### Stochastic approach to complex differential geometry and its applications

Abstract: We will discuss how stochastic differential geometry, especially coupling methods, are used in relation to the long-standing open problems of hyperbolic complex geometry, and the recent results including the stochastic Schwarz Lemma and applications. We examine further why the Kendall–Cranston probabilistic coupling method is potentially important and useful in studying differential geometry, and discusses the first Dirichlet eigenvalue estimate on Kähler manifolds, which is also a very recent result. The first part is relevant to the joint work with M. Gordina, M. Chae, and G. Yang. Also, the second part is related to the joint work with F. Baudoin, and G. Yang.

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# Past talks

## Spring 2023

### Long Time Limits of Generalized Ricci Flow

Abstract: We derive rigidity results for generalized Ricci flow blowdown limits on classes of nilpotent principal bundles. We accomplish this by constructing new functionals over the base manifold that are monotone along the flow. This overcomes a major hurdle in the nonabelian theory where the expected Perelman-type functionals were not monotone and did not yield results. Our functionals were inspired and built from subsolutions of the heat equation, which we discovered using the nilpotency of the structure group and the flow equations. We also use these and other new subsolutions to prove that, given initial data, the flow exists on the principal bundle for all positive time and satisfies type III decay bounds. In future work, we will apply these results to study the collapsing of generalized Ricci flow solutions and to classify type III pluriclosed flows on complex surfaces.

### New examples of $$\mathrm{SU} (2)^2$$-invariant $$G_2$$-instantons

Abstract: $$G_2$$-instantons are a special kind of connections on a Riemannian 7-manifold, analogues of anti-self-dual connections in 4 dimensions.
I will start this talk by giving an overview of known examples and why are we interested in them. Then, I will explain how we construct $$G_2$$-instantons in $$\mathrm{SU} (2)^2$$-invariant cohomogeneity one manifolds and give new explicit examples of $$G_2$$-instantons on $$\mathbb{R}^4 \times S^3$$ and $$S^4 \times S^3$$. I will then discuss the bubbling behaviour of sequences of $$G_2$$-instantons found.

### On the hyperbolic Bloch transform

Abstract: Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, I will introduce the noncommutative Bloch transform of Fuchsian groups, that we call the hyperbolic Bloch transform. I will prove that the hyperbolic Bloch transform is injective and "asymptotically unitary" and I will introduce a modified, geometric, Bloch transform, that transforms wave functions to sections of irreducible, flat, Hermitian vector bundles over the orbit space and transforms the hyperbolic Laplacian into the covariant Laplacian. If time permits, I will talk about potential applications to hyperbolic band theory. This is a joint work with Steve Rayan.

## Fall 2022

### Manifolds with Special Holonomy Groups and Monopole Fueter Floer Homology of 3-Manifolds

Abstract: In this talk, I propose a Floer-theoretic invariant of 3-manifolds, motivated by the study of monopoles on manifolds with special holonomy groups.

I will start with a review of the basics of manifolds with special holonomy groups and Donaldson–Segal's proposal to study these manifolds using gauge-theoretic methods. Donaldson and Segal hinted at the idea of defining invariants of Calabi–Yau 3-folds and $$\mathrm{G}_2$$-manifolds by counting monopoles on these manifolds. These monopole invariants, conjecturally, are related to the calibrated submanifolds, more specifically, special Lagrangians in Calabi–Yau 3-folds and coassociatives in $$\mathrm{G}_2$$-manifolds. This is similar to the Taubes’ theorem, which relates the Seiberg–Witten and Gromov invariants of symplectic 4-manifolds. Motivated by this conjecture, I propose numerical invariants of 3-manifolds by counting Fueter sections on hyperkähler bundles with fibers modeled on the moduli spaces of monopoles on $$\mathbb{R}^3$$. More ambitiously, one would hope this would result in a Floer-theoretic invariant of 3-manifolds. A major difficulty in defining these invariants is related to the non-compactness problems. I prove partial results in this direction, examining the different sources of non-compactness, and proving some of them, in fact, do not occur.

### H-type sub-Riemannian manifolds

Abstract: We will introduce the class of H-type sub-Riemannian manifolds. Those structures generalize H-type groups and Sasakian or 3-Sasakian structures. We will study the canonical connection on those spaces and present first eigenvalue estimates and sub-Laplacian comparison theorems.

### Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

Abstract: The fundamental gap is the difference of the first two eigenvalues of the Laplace operator, which is important both in mathematics and physics and has been extensively studied. For the Dirichlet boundary condition the log-concavity estimate of the first eigenfunction plays a crucial role, which was established for convex domains in the Euclidean space and round sphere. Joint with G. Khan, H. Nguyen and G. Wei, we obtain log-concavity estimates of the first eigenfunction for convex domains in surfaces of positive curvature and consequently establish fundamental gap estimates.

### Improved higher-order Sobolev inequalities on CR sphere

Abstract: We improve higher-order CR Sobolev inequalities on $$S^{2n+1}$$ under the vanishing of higher order moments of the volume element. As an application, we give a new and direct proof of the classification of minimizers of the CR invariant higher-order Sobolev inequalities. In the same spirit, we prove almost sharp Sobolev inequalities for GJMS operators to general CR manifolds, and obtain the existence of minimizers in $$C^{2k} (N)$$ of higher-order CR Yamabe-type problems when $$Y_k(N) < Y_k ( \mathbb{H}^n )$$.

### Another perspective on Gromov's conjectures

Abstract: For compact manifolds with boundary, to characterize the relation between scalar curvature and boundary geometry, Gromov proposed dihedral rigidity conjecture and fill-in conjecture. In this talk, we will see the role of spacetime positive mass theorem in answering the corresponding questions for initial data sets.

### Singular Weyl's Law with Ricci curvature bounded below

Abstract: The classical Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace Beltrami operator in terms of the geometry of the underlying space. Namely, the growth order is given by (half of) the dimension and the limit by the volume. The study has a long history and is important in mathematics and physics. In a very recent joint work with X. Dai, S. Honda and J. Pan, we find two surprising types of Weyl's laws for some compact Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm as some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for Ricci limit spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed by J. Pan and G. Wei (GAFA 2022).

### Applications of Geometric Flows to Questions in Nearly-Parallel $$G_2$$-Geometry

Abstract: Nearly-Parallel (NP) $$G_2$$-Structures define Einstein metrics with positive scalar curvature as well as real Killing spinors. For these reasons and more, they have a special place in differential geometry. Here we introduce the methods of geometric flows of $$G_2$$-Structures to the study of NP $$G_2$$-Structures. In particular, we consider the dynamical stability of certain NP $$G_2$$-Structures originating from 3-Sasakian geometry under the Laplacian flow and coflow. We then compare the qualitative behavior of these flows of differential forms to that of the Ricci flow for the corresponding metrics. This is joint work with Jason Lotay.

## Spring 2022

### Canonical identification between scales on Ricci-flat manifolds

Abstract: Uniqueness of tangent cone has been a central theme in many topics in geometric analysis. For complete Ricci-flat manifolds with Euclidean volume growth, the Green function for the Laplace equation can be used to define a functional which measures how fast the manifold converges to the tangent cone. If a tangent cone at infinity of the manifold has smooth cross section, Colding–Minicozzi proved that the tangent cone is unique, by showing a Łojasiewicz–Simon inequality for this functional. As an application of this inequality, we will describe how one can identify two arbitrarily far apart scales in the manifold in a natural way. We will also discuss a matrix Harnack inequality when there is an additional condition on sectional curvature, which is an elliptic analogue of matrix Harnack inequalities obtained by Hamilton and Li–Cao for geometric flows.

### Deformations of Q-curvature and generalizations

Abstract: As a fourth-order analogue, Q-curvature has similar properties as the scalar curvature in conformal geometry. Through metric deformations of Q-curvature, one can prove interesting results in Riemannian geometry such as stability and rigidity. Furthermore, we can identify the symmetric 2-tensor associated with Q-curvature, which plays the same role as how the Ricci curvature tensor associates with the scalar curvature. In this talk, my main focus is to investigate the volume comparison of Q-curvature for metrics near strictly stable Einstein metrics using variational techniques and a Morse lemma. If time permits, I will also talk about stability, rigidity and ”almost Schur lemma” of conformally variational Riemannian invariants (CVIs), which are a class of Riemannian scalar invariants satisfying similar variational properties as the scalar curvature. This talk is based on several joint works with Wei Yuan and Jeffrey Case.

### Isoperimetric properties of spaces with curvature bounded from below

Abstract: In this talk I will discuss the isoperimetric problem on spaces with curvature bounded from below. After introducing the notion of perimeter in the metric measure setting, I shall discuss the behavior of a minimizing sequence in manifolds with Ricci curvature bounded from below. In the minimization process, part of the mass might be lost at infinity in possibly non-smooth spaces. The metric measures spaces arising at infinity have Ricci bounded from below in a synthetic sense: they belong to the so-called class of RCD spaces. I will give a glance into some analytic and geometric properties of RCD spaces, and I shall give regularity results for the isoperimetric sets in RCD spaces. At the end, I will give a couple of applications. First I will show sharp differential inequalities for the isoperimetric profile in RCD spaces, second I shall present new existence criteria for the isoperimetric problem when the curvature is nonnegative. The results that I will present are new even in the smooth setting of Riemannian manifolds and exploit in a crucial way the RCD theory. The talk is based on several joint works with E. Bruè, M. Fogagnolo, S. Nardulli, E. Pasqualetto, M. Pozzetta, and D. Semola.

### On geometry of toric steady Kähler–Ricci solitons

Abstract: Let $$(M^{2n},\omega)$$ be a Kähler manifold equipped with a Hamiltonian action of a half-dimensional torus $$T^n$$. I will explain how the fundamental equations of the Kähler geometry (Kähler–Ricci flat, Kähler–Einstein and Ricci solitons) reduce to real Monge–Ampère equations for a convex function on the dual of the Lie algebra of the torus: $$Lie(T^n)^*$$. In a particular case of toric gradient steady Kähler–Ricci solitons I will prove a rigidity result showing that the only complete solitons with a free $$T^n$$ action are flat $$(C^*)^n$$. The key ingredient in this proof will be the positivity of an appropriate Bakry–Emery Ricci tensor of the orbit space $$M^{2n}/T^n$$, which — to the best of our knowledge — was not observed in the literature before.

### Gauss–Bonnet Theorems in sub-Riemannian Geometry

Abstract: The classical Gauss–Bonnet theorem is a foundational result in modern differential geometry. It relates the local notion of curvature of a manifold (which moreover depends on a choice of smooth and Riemannian structures) to global properties, in particular the (purely topological) Euler characteristic. In the context of sub-Riemannian geometry, one works with a smooth manifold equipped with an inner product defined only along a subbundle of the tangent space, and as a consequence many standard constructions from Riemannian geometry are not defined or are ill-behaved. I will present in this talk recent works establishing Gauss–Bonnet type theorems in the sub-Riemannian setting.

### Deformations of the Scalar Curvature and the Mean Curvature

Abstract: In Riemannian manifold $$(M^n, g)$$, it is well-known that its minimizing hypersurface is smooth when $$n \leqslant 7$$, and singular when $$n \geqslant 8$$. This is one of the major difficulties in generalizing many interesting results to higher dimensions, including the Riemannian Penrose inequality. In particular, in dimension 8, the minimizing hypersurface has isolated singularities, and Nathan Smale constructed a local perturbation process to smooth out the singularities. However, Smale’s perturbation will also produce a small region with possibly negative scalar curvature. In order to apply this perturbation in general relativity, we constructed a local deformation prescribing the scalar curvature and the mean curvature simultaneously. In this talk, we will discuss how the weighted function spaces help us localize the deformation in complete manifolds with boundary, assuming certain generic conditions. We will also discuss some applications of this result in general relativity.

### Non-embeddability of Carnot groups into $$L^1$$

Abstract: Motivated by the Goemans–Linial conjecture on the Sparsest Cut problem, Lee–Naor conjectured in 2006 that the Heisenberg group fails to biLipschitz embed into $$L^1$$, and this was proven true by Cheeger–Kleiner in the same year. The Heisenberg group is the simplest example of a nonabelian Carnot group, and Cheeger–Kleiner noted that their non-embeddability proof should hold for any Carnot group $$G$$ satisfying the following regularity property: For every subset $$E \subset G$$ with finite perimeter, every generic" metric-tangent space of $$E$$ at a point in $$\partial E$$ is a vertical half-space. It was expected that this property should hold for every nonabelian Carnot group, but at present, the problem remains open. The most significant achievement towards a solution is due to Ambrosio–Kleiner–Le Donne who proved that every generic" iterated metric-tangent space is a vertical-half space. In this talk, we'll describe how the result of Ambrosio–Kleiner–Le Donne together with an adaptation of the methods of Cheeger–Kleiner may be used to deduce the non-biLipschitz embedability of nonabelian Carnot groups in $$L^1$$. Based on joint work with Sylvester Eriksson–Bique, Enrico Le Donne, Lisa Naples, and Sebastiano Nicolussi–Golo.

### Singular Affine Structures, Monge–Ampère Equations and Unit Simplices

Abstract: Recent developments in complex geometry have highlighted the importance of real Monge–Ampère equations on singular affine manifolds, in particular for the SYZ conjecture concerning collapsing families of Calabi–Yau manifolds. We show that for symmetric data, the real Monge–Ampère equation on the unit simplex admits a unique Aleksandrov solution. This is concluded as a special case of a theorem giving necessary and sufficient conditions in terms of optimal transport for existence of solutions. I will outline the proof and explain a built in phenomena reminiscent of free boundary problems. Time permitting, I will discuss an application to the SYZ conjecture related to recent work by Y. Li.

### The $$L_p$$ Minkowski problem with super-critical exponents

Abstract: The classical Minkowski problem is asking for convex hypersurfaces in Euclidean space whose Gauss curvature is prescribed as a function. As an extension of the classical Minkowski problem, the $$L_p$$ Minkowski problem is asking for convex hypersurfaces with prescribed $$p$$-surface area, which is equivalent to solving a Monge–Ampère equation. In this talk, we discuss the existence of solutions to this problem for the super-critical exponents. The methods are based on the curvature flow method, some min-max ideas, and a crucial topological argument.

## Winter 2022

### Revisiting Extension Theorems in Several Complex Variables

Abstract: A key feature of several complex variables is the extension phenomenon, under pseudoconvexity or curvature conditions, starting from Hartogs’s extension of holomorphic functions, to the extension of general analytic objects such as subvarieties, coherent sheaves, meromorphic maps, etc. Kodaira’s embedding comes from the extension of jet values at points to global sections of a line bundle. Results on the Fujita conjecture and Matsusaka’s big theorem depend on effective extension results. Deformational invariance of plurigenera and analytic approaches to the finite generation of canonical ring and the abundance conjecture are based on extension techniques of $$\overline{\partial}$$ estimates from differential-geometric and PDE methods. We will discuss recent results and techniques and open problems in this area.

### Complex geometry and optimal transport

Abstract: Optimal transport studies the most economical movement of resources. In other words, one considers a pile of raw material and wants to transport it to a final configuration in a cost-efficient way. Under quite general assumptions, the solution to this problem will be induced by a transport map where the mass at each point in the initial distribution is sent to a unique point in the target distribution. In this talk, we will discuss the regularity of this transport map (i.e., whether nearby points in the first pile are sent to nearby points in the second pile). It turns out there are both local and global obstructions to establishing smoothness for the transport. When the cost is induced by a convex potential, we show that the local obstruction corresponds to the curvature of an associated Kähler manifold and discuss the geometry of this curvature tensor. In particular, we show (somewhat surprisingly) that its negativity is preserved along Kähler–Ricci flow.

### Geometric measure theory on non smooth spaces with lower Ricci curvature bounds

Abstract: The fact that locally area minimizing hypersurfaces sitting inside smooth Riemannian manifolds have vanishing mean curvature is a cornerstone of Geometric Measure Theory and of its several applications in Geometric Analysis. In this talk I will discuss how this principle can be extended and exploited on non smooth spaces with lower Ricci Curvature bounds, where the first variation formula is not available and the classical regularity theory does not even make sense.

### The first stability eigenvalues on singular hypersurfaces with constant mean curvature.

Abstract: In this talk, we study the first eigenvalue of the Jacobi operator on an integral $$n$$-varifold with constant mean curvature in space forms. We find the optimal upper bound and prove a rigidity result characterizing the case when it is attained. This gives a new characterization for certain singular Clifford tori and catenoids. The talk is based on a joint work with J.C. Pyo and Hung Tran.

### Yamabe flow of asymptotically flat metrics

Abstract: In this talk, we will discuss the behavior of the Yamabe flow on an asymptotically flat (AF) manifold. We will first show the long-time existence of the Yamabe flow starting from an AF manifold. We will then discuss the uniform estimates on manifolds with positive Yamabe constant. This would allow us to prove convergence along the Yamabe flow. If time permits, we will talk about the behavior of the rescaled flow on manifolds with negative Yamabe constant. This is joint work with Eric Chen.

### Some Remarks on Exact $$\mathit{G_2}$$-Structures on Compact Manifolds

Abstract: Closed $$\mathit{G_2}$$-Structures are utilized in the construction of every known $$\mathit{G_2}$$-Holonomy manifold, however, we have very little idea of when a compact manifold admits such a structure. Here I will survey what is known about closed $$\mathit{G_2}$$-Structures on compact manifolds and discuss my work concerning the existence of exact $$\mathit{G_2}$$-Structures on compact manifolds. Along the way I will prove several new results on soliton solutions of the Laplacian flow of $$\mathit{G_2}$$-Structures.

### Calabi–Yau Gauge Theory with Symmetries

Abstract: I will discuss the Calabi–Yau instanton and monopole equations for Calabi–Yau 3-folds, which are analogues of the classical anti-self-duality and Bogomol'nyi monopole equations in dimensions 4 and 3. I will speak about my recent work (arxiv.org/abs/2110.05439) describing the moduli-space of solutions, in the special case that the Calabi–Yau 3-fold admits a co-homogeneity one symmetry, and time permitting, report on the progress of a project to use these results to construct instantons Riemannian metrics with holonomy $$\mathit{G_2}$$.

### Asymptotics of finite energy monopoles on AC 3-manifolds

Abstract: I will report on my recent work on sharp decay estimates for critical points of the SU(2) Yang–Mills–Higgs energy functional on asymptotically conical (AC) 3-manifolds, generalizing classical results of Taubes in the 3-dimensional Euclidean space. In particular, I will explain how we prove the quadratic decay of the covariant derivative of the Higgs field of any critical point in this general context and, with an additional hypothesis on the link, we will also sketch the proof of the quadratic decay of the curvature by combining Bochner formulas with certain refined Kato inequalities with "error terms" and standard elliptic techniques. We deduce that every irreducible critical point converges along the conical end to a limiting configuration at infinity consisting of a reducible Yang–Mills connection and a parallel Higgs field. If time permits, I will mention a few open problems and future directions in this theory.

### The Heat Kernel Method in $$RCD(K,N)$$ Spaces and Its Applications to Non-collapsed Spaces

Abstract: The intrinsic notion of Non-collapsed RCD spaces was defined by Gigli–De Philippis inspired by Colding’s volume convergence theorem on Ricci limit spaces. Moreover, Colding and Naber showed that if the top dimensional density of the reference measure is finite on a set of positive measure (hence almost everywhere) then the Ricci limit space is non-collapsed. The same was conjectured to hold on RCD spaces by Gigli–De Philippis. I will talk about the background of this conjecture and some related problems, as well as the proof of this conjecture by a “geometric flow” induced by the heat kernel, and I will explain how the heat kernel come into play. This is joint work with Brena–Gigli–Honda. If time permits I will also talk about another surprising connection between the aforementioned flow and non-collapsed spaces which is joint work with Honda.

## Fall 2021

### Special Lagrangians and Lagrangian mean curvature flow

Abstract: Building on conjectures of Richard Thomas and Shing-Tung Yau, together with the definition of Bridgeland stability conditions, a recent article by Dominic Joyce proposes to use Lagrangian mean curvature flow to "decompose" certain Lagrangian submanifolds into simpler volume minimizing Lagrangians called special Lagrangians. In this talk, I will report on joint work in progress with Jason Lotay to prove parts of Dominic Joyce's program for certain symmetric hyperKähler 4-manifolds. While the existence of the so-called Bridgeland stability conditions on Fukaya categories remains an important but difficult problem related to the existence of certain abstract algebraic structures of these categories, our work provides concrete geometric interpretations for many such algebraic notions.

### Mass rigidity for asymptotically locally hyperbolic manifolds with boundary

Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to -1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang–Chruściel–Herzlich mass integrals are well-defined for it, which is a geometric invariant that essentially measure the difference from the reference manifold. In this talk, I will present a recent result with L.-H. Huang, which characterizes ALH manifolds that minimize the mass integrals. The proof uses scalar curvature deformation results for ALH manifolds with nonempty compact boundary. Specifically, we show the scalar curvature map is locally surjective among either (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. As a direct consequence, we establish the rigidity of the known positive mass theorems.

### Symmetry reduction in sub-Riemannian geometry with applications to quantum systems

Abstract: We consider a class of sub-Riemannian structures on Lie groups where the defining distribution is spanned by a set of right invariant vector fields. Such vector fields are determined by a $$K+P$$ Cartan decomposition of the corresponding Lie algebra, and, in particular, they span the $$P$$ part of the decomposition. We present a technique to calculate objects of interest in sub-Riemannian geometry such as geodesics and cut locus. The technique is based on recognizing that these problems admit a symmetry group mapping sub-Riemannian geodesics into sub-Riemannian geodesics. This group acts on the sub-Riemannian manifold properly but not freely and the associated orbit space is in general a stratified space and not a manifold. Nevertheless on the regular part of such orbit space, $$M_r$$ it is possible to define a Riemannian metric so that the Riemannian geodesics on $$M_r$$ correspond to classes of sub-Riemannian geodesics. on the original manifold. Such a symmetry reduction technique can be used not only to find sub-Riemannian geodesics but also for general problems of nonholonomic motion planning. We illustrate the technique with problems motivated by the control of quantum mechanical systems. These examples include in particular the minimum time optimal control of two level quantum systems.

### Towards hearing three-dimensional geometric structures

Abstract: The Laplace spectrum of a compact Riemannian manifold is defined to be the set of positive eigenvalues of the associated Laplace operator. Inverse spectral geometry is the study of how this set of analytic data relates to the underlying geometry of the manifold. A (compact) geometric structure defined to be a compact Riemannian manifold equipped with a locally homogeneous metric. Geometric structures played an important role in the study of two and three-dimensional geometry and topology. In dimension two, the only geometric structures are those of constant curvature and by a result of Berger, a surface of constant curvature is determined up to local isometry by its Laplace spectrum. In this work, we study the following question: "To what extent are the three-dimensional geometric structures determined by their Laplace spectra?" Among other results, we provide strong evidence that the local geometry of a three-dimensional geometric structure is determined by its Laplace spectrum, which is in stark contrast with results in higher dimensions. This is a joint work with Ben Schmidt (Michigan State University) and Craig Sutton (Dartmouth College).

### The global shape of universal covers

Abstract: If we start with a sequence of compact Riemannian manifolds $$X_n$$ shrinking to a point, take their universal covers $$\tilde{X}_n$$, and look at them from very far, how will they look like? It is well known that if there is a limiting shape $$\tilde{X}_n \to X$$, then $$X$$ is a nilpotent group with an invariant metric. On the other hand, the spaces $$\tilde{X}_n$$ are simply connected and one could (naively) expect $$X$$ to be simply connected as well. I will discuss how limits of simply connected spaces are usually simply connected and outline a proof of how in most cases $$X$$ is simply connected.

### On A Family Of Integral Operators On The Ball

Abstract: In this work, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball $$\mathbb{B}^n$$. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and prove the uniqueness of the extremal functions in the limit case using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally invariant generalization of Carleman's inequality.

### The lower bound of the integrated Carathéodory–Reiffen metric and Invariant metrics on complete noncompact Kähler manifolds

Abstract: We seek to gain progress on the following long-standing conjectures in hyperbolic complex geometry: prove that a simply connected complete Kähler manifold with negatively pinched sectional curvature is biholomorphic to a bounded domain and the Carathéodory–Reiffen metric does not vanish everywhere. As the next development of the important recent results of D. Wu and S.T. Yau in obtaining uniformly equivalence of the base Kähler metric with the Bergman metric, the Kobayashi–Royden metric, and the complete Kähler–Einstein metric in the conjecture class but missing of the Carathéodory–Reiffen metric, we provide an integrated gradient estimate of the bounded holomorphic function which becomes a quantitative lower bound of the integrated Carathéodory–Reiffen metric. Also, without requiring the negatively pinched holomorphic sectional curvature condition of the Bergman metric on an $$n$$-dimensional complete noncompact Kähler manifold, we establish the equivalence of the Bergman metric, the Kobayashi–Royden metric, and the complete Kähler–Einstein metric of negative scalar curvature under a bounded curvature condition of the Bergman metric with some reasonable conditions which also imply nonvanishing Carathédoroy–Reiffen metric.

### The Keller–Segel equations on curved planes

Abstract: The Keller–Segel equations provide a mathematical model for chemotaxis, that is the organisms (typically bacteria) in the presence of a (chemical) substance. These equations have been intensively studied on $$\mathbb{R}^n$$ with its flat metric, and the most interesting and difficult case is the planar, $$n = 2$$ one. Less is known about solutions in the presence of nonzero curvature.
In the talk, I will introduce the Keller–Segel equations in dimension 2, and then briefly recall a few relevant known facts about them. After that I will present my main results. First I prove sharp decay estimates for stationary solutions and prove that such a solution must have mass $$8 \pi$$. Some aspects of this result are novel already in the flat case. Furthermore, using a duality to the "hard" Kazdan–Warner equation on the round sphere, I prove that there are arbitrarily small perturbations of the flat metric on the plane that do not support a stationary solution to the Keller–Segel equations.
My last result is a curved version of the logarithmic Hardy–Littlewood–Sobolev inequality, which I use to prove a result that is complementary to the above ones, as it shows that the functional corresponding to the Keller–Segel equations is bounded from below only when the mass is $$8 \pi$$.