Alma L. Albujer
Title: Natural symmetries of Kaehler manifolds (joint with Jorge Alcázar and Magdalena Caballero)
Abstract: Along this talk we will study the natrual symmetries on Kaehler manifolds such as: constant holomorphic sectional curvature, locally symmetric, semisymmetric and holomorphically pseudosymmetric Kaehler manifolds. In particular, we will study some relations between the notions of holomorphic pseudosymmetry and the classical notion of pseudosymmetric Riemannian manifold proposed by Deszcz via the double secional curvatures of Deszcz. Finally, we will also study these symmetries for complex hypersurfaces in a complex space form.
Rodrigo Avalos
Title: A Q-curvature positive energy theorem and rigidity of Q-singular asymptotically Euclidean manifolds
Abstract: In this talk we will present recent results related to a notion of energy, which is associated to fourth-order gravitational theories, where it plays an analogous role to that of the classical ADM energy in the context of general relativity. We shall show that this quantity obeys a positive energy theorem with natural rigidity in the critical case of zero energy. Furthermore, we will comment on how the resulting notion of energy is deeply connected to Q-curvature, underlying positive energy theorems for the Paneitz operator as well as several rigidity phenomena associated to Q-curvature analysis. In particular, AE Q-singular manifolds turn out to be highly rigid as a consequence of the rigidity of our positive energy theorem put together with elliptic theory on appropriate weighted spaces. As a bypass, one can show how a certain fourth order analogue of the Ricci tensor retains optimal control of the decay of an AE metric.
Fidel Fernandez
Title: Geometric analysis techniques for a variational problem in (pseudo-)Finsler geometry (joint with M. Á. Javaloyes and M. Sánchez)
Abstract: We will start by motivating the problem of extending general relativity by means of (Lorentz-)Finsler geometry, both from the physical and purely mathematical viewpoints. The search for analogues to the Einstein field equation leads to a natural functional which we will vary with respect to a pseudo-Finsler metric $L$ and a nonlinear connection $N$, thus extending the Palatini formalism. After reducing the equations to the case in which $N$ is torsion-free, we will illustrate how (under a globality hypothesis on each $T_pM$) various geometric analysis tools allow one to establish properties of their solutions:
(1) In indefinite signature, a recursive argument shows that $N$ is unique if it is required to be analytic on each $T_pM$. Its lightlike geodesics coincide with those of $L$.
(2) In Lorentzian signature, the classical maximum principle implies that $N$ must be Ricci-flat. If a certain tensor invariant of $L$ vanishes, the analyticity condition of (1) can be removed.
(3) For Riemannian metrics, the knowledge of the eigenvalues of the Laplacian on the standard sphere implies that $N$ must be the Levi-Civita connection.
Leonardo García-Heveling
Title: Global hyperbolicity through the eyes of the null distance (joint with Annegret Burtscher)
Abstract: Can one encode the causal information of a spacetime in a metric space structure? In 2016, Sormani and Vega conjectured how this may be done. In this talk, I will present some recent results confirming that the answer is yes. Not only can the causal relation be encoded, but also the condition of global hyperbolicity can be characterized via completeness of the corresponding metric space.
Alfonso García-Parrado
Title: An anisotropic gravity theory (joint with E. Miguzzi)
Abstract: We study an action integral for Finsler gravity obtained by pulling back an Einstein-Cartan-like Lagrangian from the tangent bundle to the base manifold. The vacuum equations are obtained imposing stationarity with respect to any section (observer) and are well posed as they are independent of the section. They imply that in vacuum the metric is actually independent of the velocity variable so the dynamics becomes coincident with that of general relativity.
Miguel Angel Javaloyes
Title: Anisotropic tensor calculus, parallel transport and its applications
Abstract: in this talk we will focus on the concept of anisotropic connection on a manifold M, which is a connection with Christoffel symbols depending on the direction. Firstly, we will show that there is a natural way to define parallel transport using these connections by introducing the notion of observer. This is very useful to compare quantities between two points of the manifold M and, in particular, it motivates the definition of anisotropic covariant derivative of an anisotropic tensor (its components are functions on TM). As a matter of fact, when we fix an observer V, we obtain expressions for these covariant derivatives in terms of (isotropic) covariant derivatives, and all the computations can be undertaken in a free-index way. Finally, we will apply this approach:
1) to show that null-divergence of an anisotropic stress-energy tensor in General Relativity can be interpreted as local conservation of energy and momentum,
2) to obtain the fundamental equations of a Finsler submersion and,
3) to compute the flag curvature of a submanifold in a Randers-Minkowski space.
Ilyas M. Khan
Title: Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G_2-Laplacian Flow (joint with M. Haskins and A. Payne)
Abstract: Riemannian 7-manifolds with holonomy equal to the exceptional Lie group G_2 are intensely studied objects in diverse domains of mathematics and physics. One approach to understanding such manifolds is through natural flows of 3-forms called G_2-structures, the most prominent of which is Bryant’s Laplacian flow. In general, Laplacian flow is expected to encounter finite-time singularities and, as in the case of other flows, self-similar solutions should play a major role in the analysis of these singularities. In this talk, we will discuss recent joint work with M. Haskins and A. Payne in which we prove the uniqueness of asymptotically conical gradient shrinking solitons of the Laplacian flow of closed G_2 structures. We will particularly emphasize the unique difficulties that arise in the setting of Laplacian flow (in contrast to the Ricci flow, where an analogous result due to Kotschwar and Wang is well-known) and how to overcome these difficulties.
Katrin Leschke
Title: New explicit CMC cylinder and same-lobbed CMC multibubbletons (joint work with J. Cho and Y. Ogata.)
Abstract: In this talk we will discuss how to obtain new CMC cylinder via Darboux transforms of Delaunay surfaces on multiple covers. This way, we obtain explicit parameterisations of the known Delaunay bubbleton, as well as new CMC surfaces with dihedral symmmetry. As a special case we can construct same-lobbed CMC multibubbletons via Bianchi permutablity.
Steen Markvorsen
Title: A view towards some applications of Riemann-Finsler geometry
Abstract: In this talk I will comment on some Finsler-type geometric key phenomena that arise naturally in such otherwise disparate fields and topics as: Seismology; dMRI (diffusion Magnetic Resonance Imaging); Wildfire Modelling; the metric of Colour Space; and (if time permits) Riemann-Finsler conductivity.
Antonio-Martinez-Triviño
Title: A Calabi-Bernstein type Theorem for $[\varphi,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^{3}$
Abstract: Bearing in mind the Osserman-Chern point of view of the Bernstein problem for classical minimal surfaces in $\mathbb{R}^{3}$, we characterize the vertical planes prescribing the Gauss map of a properly embedded $[\varphi,\vec{e}_{3}]$-minimal surface in $\mathbb{R}^{3}$ whose Gauss curvature is non-negative under some analytical conditions over the function $\varphi$. Furthermore, as a consequence of this result together with a Calabi type correspondence between these surfaces and its corresponding spacelike surfaces in the Lorentz-Minkowski space $\mathbb{L}^{3}$, we give a non-existence result for complete $[\varphi,\vec{e}_{3}]$-maximal surfaces whose Gauss map lies in a closed region of the anti-De Sitter space.
Niels M. Moller
Title: On the entropy spectrum of embedded self-shrinkers with symmetries (joint with Ali Muhammad and John Ma)
Abstract: We consider complete shrinking solitons for the mean curvature flow of hypersurfaces in Rn+1. Using comparison geometry, we find an explicit universal constant bounding the entropies of all such embedded self-shrinkers with rotational symmetry. As applications, we prove within this class smooth compactness and, via the Lojasiewicz-Simon inequality, some finiteness theorems extending previous results obtained by Mramor.
David Moya
Título: Index of a plane through the origin in the k-Schwarzschild space (joint with Ezequiel Barbosa)
Abstract: In this talk we show the existance of a family of properly embedded free boundary minimal surfaces of revolution with circular boundaries in the horizon of the $k$-Schwarzschild space, which answers a question proposed by O. Chodosh and D. Ketover on the existence of non-totally geodesic minimal surfaces in the 3-dimensional Schwarzschild space. This one parameter family of surfaces converges to $\Sigma_0=\{x\in \mathbb{R}^n; x_n=0, |x|\geq R_0\}.$ It was proved by R.Montezuma that the Morse index of $\Sigma_0$ in the Schwarzschild space is one. We generalized this result to compute the Morse index of $\Sigma_0$ in the $k$-Schwarzschild space for particular situations.
Alexander E. Mramor
Title: Some applications of the mean curvature flow to self shrinkers
Abstract: In this title I'll discuss some applications of the mean curvature flow to self shrinkers in R3 and R4.
Gregorio Pacelli Bessa
Title: Green Functions and the Dirichlet Spectrum (joint with V. Gimeno and L. Jorge)
Abstract: I will talk about the Green operator and the Dirichlet spectrum of bounded domains of Riemannian manifolds. More precisely, I will talk about four types of results:
(1) The iteration of the Green operator due to Sadao Sato to obtain the first Dirichlet eigen- value of bounded domains
(2) The L1-moment spectrum, the results here are related to the work of McDonald-Meyers and Hurtado-Markvorsen-Palmer.
(3) The radial spectrum of balls centred at the origin of model manifolds.
(4) Extrinsic volume of properly immersed minimal submanifolds and the Dirichlet spectrum.
Enrique Pendás
Title: Zermelo's problem and its application to wave propagation
Abstract: Zermelo's problem seeks the fastest trajectory between two prescribed points for a moving object in the presence of a current. In the most general case, where the velocity of the object may depend on the space, time and direction, the solution to this problem can be interpreted as a lightlike geodesic of a certain Finsler spacetime. This can be translated to waves: lightlike geodesics orthogonal to the initial wavefront are the trajectories that minimize the propagation time and, therefore, make up the wavefront. Furthermore, this theoretical framework can be applied to any physical phenomenon that satisfies Huygens' principle (i.e., behaves as a wave in terms of propagation), such as the spread of wildfires.
Julián Pozuelo
Title: The prescribed mean curvature equation for $t$-graphs in the Sub-Finsler Heisenberg group $\mathbb{H}^n$ (joint with Giovannardi, Pinamonti and Verzellesi)
Abstract: In the Heisenberg group $\mathbb{H}^n$ we consider a notion of perimeter associated to a convex body $K\subset \mathbb{R}^{2n}$ of class $C^2_+$ containing $0$ in its interior. By means of the first variation formula, we define a natural notion of sub-Finsler mean curvature of a $t$-graph and study the sub-Finsler prescribed mean curvature equation on a bounded domain $\Omega$. When the prescribed datum $H$ is constant and strictly smaller than the anisotropic mean curvature of $\partial \Omega$, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of an approximation scheme.
Mariel Sáez
Title: Uniqueness of entire graphs evolving by mean curvature flow (joint with P. Daskalopoulos)
Abstract: In this work we study the uniqueness of graphical mean curvature flow with locally Lipschitz initial data. We first prove that rotationally symmetric entire graphs are unique, without any further assumptions. Our methods also give an alternative simple proof of uniqueness in the one dimensional case. In the general case, we establish the uniqueness of entire proper graphs that satisfy a uniform lower bound on the second fundamental form. The latter result extends to initial conditions that are proper graphs over subdomains of Rn. A consequence of our result is the uniqueness of convex entire graphs, which allows us to prove that Hamilton’s Harnack estimate holds for mean curvature flow solutions that are convex entire graphs.
Handan Yildirim
Title: Rotationally Invariant Translating Solitons in Einstein's Space-time (joint with Miguel Ortega)
Abstract: In this talk which is based on a joint work with Miguel Ortega, non-degenerate SO(n)-invariant translating solitons of the mean curvature flow in Einstein's Space-time travelling in the time-like real direction are classified obtaining three space-like families (any of which is of Type either I, II or III) and four time-like families (any of which is of Type either IV, V, VI or VII). Each example has one or two conic singularities. Moreover, two tangency principles are stated for general space-like translating solitons which are not necessarily SO(n)-invariant. As an application, it is proved that if a compact space-like translating soliton has no singularities and its boundary is contained in a slice as a small sphere, then it is a compact piece of an example of Type I.
The speaker would like to thank the Spanish MICINN for the financial support through the project PID2020-116126GB-I00 in which she is involved in the working team as a researcher. She alsowould like to thank the Research Group FQM-324 of the Geometry and Topology Department of Granada University for the financial support provided during her visit in July 2022 since this talk is related to the work initiated in that period.