Scientific program

• Ana Menezes (Princeton University)

Title: Eigenvalue problems and free boundary minimal surfaces in spherical caps.

Abstract: In a joint work with Vanderson Lima (UFRGS, Brazil), we introduced a family of functionals on a space of Riemannian metrics of a compact surface with boundary, defined via eigenvalues of a Steklov-type problem. In this talk we will prove that each such functional is uniformly bounded from above, and we will characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we will prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions.

• Anna Fino (Università di Torino)

Title: Strong HKT geometry on hypercomplex manifolds.

Abstract: A hyperhermitian manifold (M, J₁, J₂, J₃, g) is said to be strong HKT if the Bismut connections associated to the three Hermitian structures (Jᵢ, g), for i = 1, 2, 3, coincide, and the common Bismut torsion 3-form is closed.

In the  talk  I  will  first discuss  some general properties of strong HKT manifolds, including a non-existence result on solvmanifolds.   Then I  will  focus on  the geometry of compact, simply connected strong HKT manifolds in real dimension eight.  This talk  is based on a  joint work with Beatrice Brienza,  Gueo  Grantcharov, and Misha Verbitsky.

• Francesca Oronzio (Scuola Superiore Meridionale)

Title: ADM Mass and Potential Theory.

Abstact: In this talk, we describe some monotonicity formulas that hold along the regular level sets of suitable p–harmonic functions in asymptotically flat Riemannian 3–manifolds with a single end, with or without boundary, and with nonnegative scalar curvature. Using these formulas, we derive some geometric inequalities, including the positive mass inequality and the Riemannian Penrose inequality. 

• Jason Lotay (University of Oxford)

Title: Einstein manifolds, G₂ geometry and geometric flows.

Abstract: There are infinitely many positive Einstein metrics in dimension 7 and many can be encoded through G₂ geometry. It is natural to ask about the stability of the associated G₂ geometry with respect to natural geometric flows of G₂ structures. I will report on joint works with A. Kennon and J. Stein that address this question, demonstrating cases of both stability and instability.

• Jorge Lauret (Universidad Nacional de Córdoba)

Title: Hermitian geometry: the compact and homogeneous case.

Abstract: Starting from a flag manifold F=G/H (the only compact homogeneous manifolds which are Kähler), each of the closed tori T in the center Z(H) of even codimension defines a torus bundle M=G/K over F with fibre A=Z(H)/T, where K=[H,H]xT.  The different slopes of T in Z(H) may or may not have topological consequences on M.  These so-called C-spaces M=G/K are precisely the compact homogeneous spaces admitting invariant complex structures, which are all given by one of the finitely many complex structures on F and any left-invariant complex structure on the torus A, i.e., any linear map J_a on the Lie algebra a of A whose square is -I.  

The freedom to choose J_a is overwhelming.  In this talk, we will show that the existence of distinguished Hermitian metrics like Hermite-Einstein, balanced, SKT, CYT, Chern-Einstein, LCK, etc., can be very sensitive to such a choice.

• Lino Grama (Universidade Estadual de Campinas)

Title: TBA.

Abstract: TBA.

• Mariel Sáez (Pontificia Universidad Católica de Chile)

Title: Geometric Flows and some applications.

Abstract: Geometric flows are a fundamental tool in geometric analysis, with profound applications across both pure and applied mathematics. They have played a key role in landmark achievements, such as the resolution of the Poincaré conjecture and a proof of the Penrose inequality in general relativity.

In this talk, I will introduce several notable geometric flows, highlight their major applications, and discuss current open problems in the field.

• Megan Kerr (Wellesley College)

Title: Compact Homogeneous Einstein Manifolds.

Abstract: We consider the question of which compact homogeneous spaces M ≅ G/H admit a G-invariant Einstein metric. Einstein metrics have a variational characterization: they are exactly the critical points of the scalar curvature functional s(g) on unit volume metrics.

By work of C. Böhm (2004) , there is a simplicial complex Δ_{G/H} that detects when G-invariant Einstein metrics must exist for global reasons: on a compact homogeneous space G/H, if Δ_{G/H} is non-contractible, then G/H admits a G-invariant Einstein metric. However, there are many compact homogeneous spaces with a contractible Δ_{G/H}, which nevertheless admit one or more G-invariant Einstein metrics.

Let 𝒩^c_{0, k} the class of compact homogeneous manifolds G/H such that G is a compact, connected, simple Lie group, (i) rk(G)-rk(H)=0, (ii) the simplicial complex Δ_{G/H} is contractible (Einstein metrics are not guaranteed), and (iii) the isotropy representation of G/H has exactly k irreducible summands. In the cases where G/H has k =1 or 2 irreducible summands in the isotropy representation, we have an algebraic marker that corresponds precisely to existence of invariant Einstein metric. This work in progress is to classify the homogeneous spaces in 𝒩^c_{0, k} for k ≤ 4, and to determine which such spaces admit at least one G-invariant Einstein metric. Ultimately, we want to identify more algebraic tools for k>2.

• Nicoletta Tardini (Università di Parma)

Title: TBA.

Abstract: TBA.

• Ryuichi Fukuoka (Universidade Estadual de Maringá)

Title: Smoothing of left-invariant strongly convex C⁰-Finsler structures on Lie groups and convergence of extremals.

Abstract: A C⁰-Finsler structure on a smooth manifold M is a continuous function F : TM → R such that F restricted to each tangent space is an asymmetric norm. F is strongly convex if its restriction to each tangent space is a strongly convex asymmetric norm. Let G be a Lie group endowed with a left-invariant strongly convex C⁰-Finsler structure F. In this work we introduce a smoothing Fε of F, which is a one-parameter family of Finsler structures converging to F. This is a left invariant version of the smoothing introduced by the same authors in [1]. We study extremals x(t) on (G,F) using the Pontryagin maximum principle. Given (x₀,α₀) in the contangent bundle T*G of G, we prove that there exist a unique Pontryagin extremal t ∈ R↦ (x(t), α(t)) ∈ T*G such that (x(0), α(0)) = (x₀,α₀). Moreover, if t ∈ R ↦ (xε(t), αε(t)) is the unique Pontryagin extremal in (G, Fε) such that (xε(0), αε(0)) = (x₀,α₀), then we prove that the family of extremals xε(t) restricted to compact intervals of R converges uniformly to the extremal x(t) as ε → 0. This is a joint work with Anderson Macedo Setti.

References

[1] Fukuoka, R.; Setti, A. M. Mollifier smoothing of C⁰-Finsler structures. Ann. Mat. Pura Appl. 200 (2021) 595-639.