Mean curvature flow is the gradient flow of the area functional where an
embedded hypersurface evolves in direction of its mean curvature vector. This
constitutes a natural geometric heat equation for hypersurfaces, which ideally will
evolve the embedding into a nicer shape. But due to the nonlinear nature of the
equation singularities are guaranteed to form. Nevertheless, a key...
Τhe affine translational group, that is HXR, where H is the hyperbolic plane and R the set of the real numbers, endowed with a natural product, is one of the Thurston 3-geometries. It turns out that, like the Heisenberg group, a theory of quasiconformal mappings may be developed for this group. We briefly discuss the main features of the theory. This is a joint work in progress together with...
Higher Teichmüller theory pertains to the study of special connected components of character varieties sharing analogous properties to the classical Teichmüller space. By fixing a complex structure on the underlying topological surface, one introduces powerful holomorphic techniques via Higgs bundles, the latter corresponding to fundamental group representations through the non-abelian Hodge...
The Atiyah class is an obstruction class for the existence of various geometric structures depending on the context. The original construction of Atiyah measures the obstruction to existence of holomorphic connections on holomorphic vector bundles. Since then the class has seen many applications and generalizations ranging from representation theory to quantum field theory. We construct the...
In this talk we will present some results concerning the Fried's conjecture, i.e., the relation of the twisted dynamical zeta function of Ruelle at zero and spectral invariants for a hyperbolic manifold X. In particular, we consider the twisted Ruelle zeta function twisted by an arbitrary representation of the lattice. We study then its relation to the Ray-Singer norm of the refined analytic...
In his groundbreaking paper, "Supersymmetry and Morse theory", Witten categorified the classical Morse polynomials by constructing instanton complexes, and extended them to Dolbeault complexes on spaces with Kahler Hamiltonian actions in a follow up article, generalizing Lefschetz-Riemann-Roch formulas. We recently extended these results to stratified pseudomanifolds with wedge (asymptotically...
In this talk we will present some recent results concerning the $L^p$-spectrum of the Laplacian on $k$-forms. We will generalize the set of manifolds over which the $L^p$-spectrum of the Laplacian on $k$-forms depends on $p$. We will consider the case of manifolds that are warped products at infinity and certain quotients of Hyperbolic space. In the case of warped products at infinity we will...
In the analysis of variational problems it is often important to understand not only the behaviour of exact minimisers and critical points, but also of maps that almost minimise the energy or that almost solve the associated Euler-Lagrange equation.
It is in particular natural to ask whether the distance of an almost minimiser to the nearest minimising state is controlled in terms of the...
The linking number of two disjoint oriented null-homologous submanifolds K,L in an oriented manfold M is the primary obstruction to disentangling them. By means of the Thom isomorphism in deRham theory, the linking number can be rewritten as an integral over the product of the two submanfolds, analogous to Gauss' famous linking integral for disjoint loops in euclidean space.
The integrand...
We study locally harmonic maps between pseudo-Riemannian surfaces.
These maps are classified by the classification of the solutions of a
generalized sine-Gordon equation. We then study the one-soliton solutions
of this equation and we find the corresponding harmonic maps in a unified
way. Finally, we discuss a Backlund transformation of the harmonic map
equations that provides a...
We study harmonic maps from a subset of the complex plane to a subset of the hyperbolic plane. In Fotiadis and Daskaloyannis (Nonlinear Anal 214, 112546, 2022), harmonic maps are related to the sinh-Gordon equation and a Bäcklund transformation is introduced, which connects solutions of the sinh-Gordon and sine-Gordon equation. We develop this machinery in order to construct new harmonic maps...
It is known that constant mean curvature (CMC) hypersurfaces, with or without boundary, in a smooth Riemannian manifold $(M, g)$ can develop singularities if $n\geq 7$. For a CMC hypersurface $S$ that minimises the prescribed mean curvature functional in a ball $B_r(0)$ with boundary $\partial S$ in $\partial B_r(0)$ and an isolated singularity at the origin we construct in [enter link...
In a previous study of ours we considered two concentric conics $C_1,C_2$ intersecting at four points and we searched all conics having double contact with these two. As a solution we found an one-parameter family of conics, the so-called double contact conics of $C_1,C_2$. We noticed that this family creates a hyperbolic involution $f_{AB}$ on the pencil of lines through their common centre...
In this talk we establish a gradient bound and Liouville-type
theorems for solutions to Quasi-linear equations on compact Riemannian
manifolds with nonnegative Ricci curvature. Moreover, we obtain a
Harnack-type inequality and an Alexandrov-Bekelman-Pucci type estimate for
the gradient of solutions in domains contained in the manifold.
(This is joint work with George Zacharopoulos, EKPA)
Our goal is to show how to improve some results related to the title of this talk, due to Bianchini, Colesanti, Pagnini and Roncoroni. Namely, we prove the log-Brunn-Minkowski inequality for intrinsic volumes (in fact the $L_p$-Brunn-Minkowski inequality for negative $p$) in a $C^2$ neighbourhood of the euclidean ball. On the other hand, we show that the $L_p$-Brunn-Minkowski inequality for...
We present quantitative stability aspects of the class of Möbius transformations of $\mathbb{S}^{n-1}$ among maps in the critical Sobolev space $W^{1,n-1}$. The case of $\mathbb{S}^{n-1}$- and general $\mathbb{R}^n$-valued maps will be addressed. In the latter, more flexible setting, unlike similar in flavour results for maps defined on domains of $\mathbb{R}^n$, not only a conformal deficit...
