We present a new construction of embedded minimal surfaces in hyperbolic space with 3 asymptotically totally geodesic ends and arbitrary finite genus.

We describe the structure of the singular sets of constant curvature, convex hypersurfaces in hyperbolic space for general convex curvature functions. We apply this result to the study of the ideal Plateau problem in hyperbolic space for such curvature functions.

In [17], Labourie initiated the study of the dynamical properties of the space of $k$-surfaces, that is, suitably complete immersed surfaces of constant extrinsic curvature in $3$-dimensional manifolds, which he presented as a higher-dimensional analogue of the geodesic flow when the ambient manifold is negatively curved. In this paper, following the recent work [5] of Calegari--Marques--Neves, we study the asymptotic counting of surface subgroups in terms of areas of $k$-surfaces. We determine a lower bound, and we prove rigidity when this bound is achieved. Our work differs from that of [5] in two key respects. Firstly, we work with all quasi-Fuchsian subgroups as opposed to merely asymptotically Fuchsian ones. Secondly, as the proof of rigidity in [5] breaks down in the present case, we require a different approach. Following ideas outlined by Labourie in [19], we prove rigidity by solving a general foliated Plateau problem in Cartan--Hadamard manifolds. To this end, we build on Labourie's theory of $k$-surface dynamics, and propose a number of new constructions, conjectures and questions.

Labourie raised the question of determining the possible asymptotics for the growth rate of compact k-surfaces, counted according to energy, in negatively curved 3-manifolds, indicating the possibility of a theory of thermodynamical formalism for this class of surfaces. Motivated by this question and by analogous results for the geodesic flow, we prove a number of results concerning the asymptotic behavior of high energy k-surfaces, especially in relation to the curvature of the ambient space. First, we determine a rigid upper bound for the growth rate of quasi-Fuchsian k-surfaces, counted according to energy, and with asymptotically round limit set, subject to a lower bound on the sectional curvature of the ambient space. We also study the marked energy spectrum for k-surfaces, proving a number of domination and rigidity theorems in this context. Finally, we show that the marked area and energy spectra for k-surfaces in 3-dimensional manifolds of negative curvature are asymptotic if and only if the sectional curvature is constant.

For 0 < k < 1, a finite-type k-surface in 3-dimensional hyperbolic space is a complete, immersed surface of finite area and of constant extrinsic curvature equal to k. In Smith (2021), we showed that such surfaces have finite genus and finitely many cusp-like ends. Each of these cusps is asymptotic to an immersed cylinder of exponentially decaying radius about a complete geodesic and terminates at an ideal point which we call the extremity of the cusp. We show that every cusp of any finite-type k-surface has a well-defined axis, which we will call the Steiner geodesic of the cusp. One of the end-points of this axis is the extremity, and we will call the other, which constitutes new geometric data, the Steiner point of the cusp. We prove a new identity involving extremities and Steiner points in terms of Mobius invariant vector fields over the Riemann sphere. We define two new functionals over the space of finite-type k-surfaces. The first, which will be called the generalized volume, is defined by the integral of a certain well-chosen form, and extends to the non-embedded case the concept of volume of the set bounded by the surface. The second, which will be called the renormalized energy, is related to the integral of the mean curvature of the surface, and is well-defined up to a choice of Busemann function. Upon describing natural parametrizations of the strata of the space of finite-type k-surfaces by open complex manifolds, we prove a new Schlafli-type formula relating the extremities and Steiner points to the first order variations of the generalized volume and the renormalized energy. In particular, Mobius invariance of this formula yields the aforementioned identity. We conclude by studying some applications of this identity and Schlafli-type formula.

We desingularise the union of $3$ Grim paraboloids along Costa-Hoffman-Meeks surfaces in order to obtain complete embedded translating solitons of the mean curvature flow with $3$ ends and arbitrary finite genus.

In the study of immersed surfaces of constant positive extrinsic curvature in space-forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such surfaces under the hypotheses of quasicompleteness. In particular, we show that, for k>Max(0,-c), the only quasicomplete immersed surfaces of constant extrinsic curvature equal to k in the 3-dimensional space-form of constant sectional curvature equal to c are the geodesic spheres. Together with earlier work of the author, this completes the classification of quasicomplete immersed surfaces of constant positive extrinsic curvature in 3-dimensional space-forms.

We use Clifford algebras to construct a unified formalism for studying constant extrinsic curvature immersed surfaces in Riemannian and semi-Riemannian 3-manifolds in terms of immersed bilegendrian surfaces in their unitary bundles. As an application, we provide full classifications of both complete and compact immersed bilegendrian surfaces in the unit tangent bundle U𝕊^3 of the 3-sphere.

We construct eternal mean curvature flows of tori in perturbations of the standard unit sphere S-3. This has applications to the study of the Morse homologies of area functionals over the space of embedded tori in S-3.

We present a basic introduction to the theories of Möbius structures and hyperbolic ends and we study their applications to the theory of $k$-surfaces in $3$-dimensional hyperbolic space.

In the present paper, we revisit a famous theorem by Candel that we generalize by proving that given a compact lamination by hyperbolic surfaces, every negative function smooth inside the leaves and transversally continuous is the curvature function of a unique laminated metric in the corresponding conformal class. We give an interpretation of this result as a continuity result about the solutions of some elliptic PDEs in the so called Cheeger-Gromov topology on the space of complete pointed riemannian manifolds.

Let $S$ be a closed surface of hyperbolic type. We show that, for every pair $(g_+,g_-)$ of negatively curved metrics over $S$ there exists a unique GHMC Minkowski spacetime $X$ into which $(S,g_+)$ and $(S,g_-)$ isometrically embed as Cauchy surfaces in the future and past components respectively.

We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichm\"uller theory than arbitrary non-compact surfaces. We show that the Teichm\"uller space of any non-trivial hyperbolic surface lamination is infinite dimensional. In order to prove this result, we study the theory of deformations of hyperbolic surfaces, and we derive what we believe to be a new formula for the derivative of the length of a simple closed geodesic with respect to the action of grafting. This formula complements those derived by McMullen in [23], in terms of the Weil-Petersson metric, and by Wolpert in [33], for the case of earthquakes.

In the present paper, we revisit a famous theorem by Candel that we generalize by proving that given a compact lamination by hyperbolic surfaces, every negative function smooth inside the leaves and transversally continuous is the curvature function of a unique laminated metric in the corresponding conformal class. We give an interpretation of this result as a continuity result about the solutions of some elliptic PDEs in the so called Cheeger-Gromov topology on the space of complete pointed riemannian manifolds.

We prove dynamical stability of a natural class of hypersurface laminations defined over Cartan–Hadamard manifolds of pinched curvature. We achieve this by providing a complete solution to the asymptotic Plateau problem for immersed surfaces of constant extrinsic curvature in Cartan–Hadamard manifolds, proposed by Labourie in [25], together with its natural higher-dimensional generalisation.

For all n, we define the n-dimensional critical catenoid $$M_n$$ to be the unique rotationally symmetric, free boundary minimal hypersurface of non-trivial topology embedded in the closed unit ball in $$\mathbb {R}^{n+1}$$ . We show that the Morse index $$\mathrm{MI}(n)$$ of $$M_n$$ satisfies the following asymptotic estimate as n tends to infinity. $$\begin{aligned} \mathop {{\text {Lim}}}_{n\rightarrow +\infty }\frac{\text {Log}(\mathrm{MI}(n))}{\sqrt{n}\text {Log}(\sqrt{n})} = 1. \end{aligned}$$ We illustrate our results with an in-depth study of the numerical problem, providing exact values for the Morse index for $$n=2,\ldots ,100$$ , together with qualitative studies of $$\mathrm{MI}(n)$$ and related geometric quantities for large values of n.

Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a functional introduced by Bonsante, Mondello \& Schlenker, we show that there exists a unique affine deformation $\Gamma:=(\rho,\tau)$ of a Fuchsian group such that $(S,g_+)$ and $(S, g_-)$ embed isometrically as locally strictly convex Cauchy surfaces in the future and past complete components respectively of the quotient by $\Gamma$ of an open subset $\Omega$ of Minkowski space. Such quotients are known as Globally Hyperbolic, Maximal, Cauchy compact Min\-kow\-ski spacetimes and are naturally dual to the half-pipe spaces introduced by Danciger. When translated into this latter framework, our result states that there exists a unique, marked, quasi-Fuchsian half-pipe space in which $(S, g_+)$ and $(S, g_-)$ are realised as the third fundamental forms of future- and past-oriented, locally strictly convex graphs.

We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for LSC hypersurfaces of constant or prescribed curvature for general curvature functions inside general Hadamard manifolds modulo a single scalar condition. In particular, convex curvature functions of bounded type are fully treated.

Let (M,Q) be a compact, three dimensional manifold of strictly negative sectional curvature. Let (Σ, P ) be a compact, orientable surface of hyperbolic type (i.e. of genus at least two). Let θ : π1(Σ, P ) → π1(M,Q) be a homomorphism. Generalising a recent result of Gallo, Kapovich and Marden concerning necessary and sufficient conditions for the existence of complex projective structures with specified holonomy to manifolds of non-constant negative curvature, we obtain necessary conditions on θ for the existence of a so called θ-equivariant Plateau problem over Σ, which is equivalent to the existence of a strictly convex immersion i : Σ →M which realises θ (i.e. such that θ = i∗).

We develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. We apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where $K$ is mean curvature; extrinsic curvature and special Lagrangian curvature, and we show that in all these cases, this number is equal to $-\chi(M)$, where $\chi(M)$ is the Euler characteristic of $M$.