Publications

Preprints

• José S. Santiago, Uniqueness of the Bonnet problem in Thurston geometries. arXiv:2512.01602 [math.DG]
  Abstract

  We study the Bonnet problem in Bianchi–Cartan–Vrănceanu spaces and in $\mathrm{Sol}_3$. Our main contribution is to establish the uniqueness of Bonnet mates, which leads us to address the problem of determining when an isometric immersion can be continuously deformed through isometric immersions that preserve the principal curvatures – a question originally posed in $\mathbb{R}^3$ by Chern~\cite{Chern}. For Bianchi–Cartan–Vrănceanu spaces, we complete the local classification of Bonnet pairs by studying the uniqueness of the results obtained by Gálvez, Martínez and Mira [GMM], and we provide new examples of Bonnet mates that were not previously considered. In particular, we prove that the aforesaid continuous deformations only exist for minimal surfaces in the product spaces $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$ and otherwise only for surfaces with constant principal curvatures. In the case of $\mathrm{Sol}_3$, we give a characterization of Bonnet mates via a system of two differential equations, addressing a problem proposed in [GMM]. We conclude that the only surfaces admitting continuous isometric deformations that preserve the principal curvatures in $\mathrm{Sol}_3$ are those with constant left-invariant Gauss map.

• Ildefonso Castro, José M. Manzano, & José S. Santiago, Isometric immersions into three-dimensional unimodular metric Lie groups. arXiv:2507.16728 [math.DG]
  Abstract

  We study isometric immersions of surfaces into simply connected 3-dimensional unimodular Lie groups endowed with either Riemannian or Lorentzian left-invariant metrics, assuming that Milnor’s operator is diagonalizable in the Lorentzian case. We provide global models in coordinates for all these metric Lie groups that depend analytically on the structure constants and establish some fundamental theorems characterizing such immersions. In this sense, we study up to what extent we can recover the immersion from (a) the tangent projections of the natural left-invariant ambient frame, (b) the left-invariant Gauss map, and (c) the shape operator. As an application, we prove that an isometric immersion is determined by its left-invariant Gauss map up to certain well controlled angular companions. We also we classify totally geodesic surfaces and introduce four Lorentzian analogues of the Daniel correspondence within two families of Lorentzian homogeneous 3-manifolds with 4-dimensional isometry group. We also classify isometric immersions in $\mathbb{R}^3$ or $\mathbb{S}^3$ whose left-invariant Gauss maps differ by a direct isometry of $\mathbb{S}^2$. Finally, we show that Daniel’s is the furthest extension of the classical Lawson correspondence for constant mean curvature surfaces within Riemannian unimodular metric Lie groups.

• Andrea Del Prete, José M. Manzano, & Barbara Nelli, The Jenkins-Serrin problem in 3-manifolds with a Killing vector field. arXiv:2306.12195 [math.DG]
  Abstract

  We consider a Riemannian submersion from a 3-manifold $\mathbb{E}$ to a surface $M$, both connected and orientable, whose fibers are the integral curves of a Killing vector field without zeros, not necessarily unitary. We solve the Jenkins-Serrin problem for the minimal surface equation in $\mathbb{E}$ over a relatively compact open domain $\Omega\subset M$ with prescribed finite or infinite values on some arcs of the boundary under the only assumption that the same value $+\infty$ or $-\infty$ cannot be prescribed on two adjacent components of $\partial\Omega$ forming a convex angle. The domain $\Omega$ can have reentrant corners as well as closed curves in its boundary. We show that the solution exists if and only if some generalized Jenkins-Serrin conditions (in terms of a conformal metric in $M$) are fulfilled. We develop further the theory of divergence lines to study the convergence of a sequence of minimal graphs. We also provide maximum principles that guarantee the uniqueness of the solution. Finally, we obtain new examples of minimal surfaces in $\mathbb{R}^3$ and in other homogeneous $3$-manifolds.

Peer-reviewed published papers

• Paula Carretero, Ildefonso Castro, Rotational surfaces with prescribed curvatures. Diff. Geom. Appl., (101), 2025, 1—19.
  10.1016/j.difgeo.2025.102298. arXiv:2312.14672
  Abstract

  We solve the problem of prescribing different types of curvatures (principal, mean or Gaussian) on rotational surfaces in terms of arbitrary continuous functions depending on the distance from the surface to the axis of revolution. In this line, we get the complete explicit classification of the rotational surfaces with mean or Gauss curvature inversely proportional to the distance from the surface to the axis of revolution. We also provide new uniqueness results on some well known surfaces, such as the catenoid or the torus of revolution, and others less well known but equally interesting for their physical applications, such as the Mylar balloon or the Flamm’s paraboloid.

• Andrea Del Prete, Minimal graphs over non-compact domains in 3-manifolds with a Killing vector field. Results Math, 80(192), 2025, 1—32.
  10.1007/s00025-025-02500-8. arXiv:2307.09240
  Abstract

  Let $\mathbb{E}$ be a connected and orientable Riemannian 3-manifold with a non-singular Killing vector field whose associated one-parameter group of the isometries of $\mathbb{E}$ acts freely and properly on $\mathbb{E}$. Then, there exists a Killing Submersion from $\mathbb{E}$ onto a connected and orientable surface $M$ whose fibers are the integral curves of the Killing vector field. In this setting, assuming that $M$ is non-compact and the fibers have infinite length, we solve the Dirichlet problem for minimal Killing graphs over certain unbounded domains of $M$, prescribing piecewise continuous boundary values. We obtain general Collin-Krust type estimates. In the particular case of the Heisenberg group, we prove a uniqueness result for minimal Killing graphs with bounded boundary values over a strip. We also prove that isolated singularities of Killing graphs with prescribed mean curvature are removable.

• Jesús Castro-Infantes, José S. Santiago, Genus one $H$-surfaces with $k$-ends in $\mathbb{H}^2\times \mathbb{R}$. Rev. Mat. Iberoam., 41(1), 2025, 365—400.
  10.4171/RMI/1536. arXiv:2306.17433
  Abstract

  We construct two different families of properly Alexandrov-immersed surfaces in $\mathbb{H}^2\times \mathbb{R}$ with constant mean curvature $0<H\leq \frac 1 2$, genus one and $k\geq2$ ends ($k=2$ only for one of these families). These ends are asymptotic to vertical $H$-cylinders for $0<H<\frac 1 2$. This shows that there is not a Schoen-type theorem for immersed surfaces with positive constant mean curvature in $\mathbb{H}^2\times\mathbb{R}$. These surfaces are obtained by means of a conjugate construction.

• Ildefonso Castro, Ildefonso Castro-Infantes, & Jesús Castro-Infantes, Helicoidal minimal surfaces in the 3-sphere: An approach via spherical curves. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat, 118(77), 2024, 1—21.
  10.1007/s13398-024-01574-3. arXiv:2303.08043
  Abstract

  We prove an existence and uniqueness theorem about spherical helicoidal (in particular, rotational) surfaces with prescribed mean or Gaussian curvature in terms of a continuous function depending on the distance to its axis. As an application in the case of vanishing mean curvature, it is shown that the well-known conjugation between the belicoid and the catenoid in Euclidean three-space extends naturally to the 3-sphere to their spherical versions and determine in a quite explicit way their associated surfaces in the sense of Lawson. As a key tool, we use the notion of spherical angular momentum of the spherical curves that play the role of profile curves of the minimal helicoidal surfaces in the 3-sphere.

• Andrea Del Prete, Vincent Gimeno, Parabolicity of Invariant Surfaces. The Journal of Geometric Analysis, 34(109), 2024, 1—15.
  10.1007/s12220-024-01552-6. arXiv:2312.02837
  Abstract

  We present a clear and practical way to characterize the parabolicity of a complete immersed surface that is invariant with respect to a Killing vector field of the ambient space.

• Andrea Del Prete, Hojoo Lee, & José M. Manzano, A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions. Math. Nachrichten, 297(5), 2024, 1581—1600.
  10.1002/mana.202300282. arXiv:2306.00562
  Abstract

  We develop a conformal duality for space-like graphs in Riemannian and Lorentzian three-manifolds that admit a Riemannian submersion over a Riemannian surface whose fibers are the integral curves of a Killing vector field, which is time-like in the Lorentzian case. The duality swaps mean curvature and bundle curvature and sends the length of the Killing vector field to its reciprocal while keeping invariant the base surface. We obtain two consequences of this result. On the one hand, we find entire graphs in Lorentz–Minkowski space $\mathbb{L}^3$ with prescribed mean curvature a bounded function $H \in \mathcal{C}^\infty(\mathbb{R}^2)$ with bounded gradient. On the other hand, we obtain conditions for the existence and nonexistence of entire graphs which are related to a notion of the critical mean curvature.

• José M. Manzano, Invariant constant mean curvature tubes around a horizontal geodesic in $\mathbb{E}(\kappa, \tau)$-spaces. J. Math. Appl., 531(1 (Part 2)), 2024, —.
  10.1016/j.jmaa.2023.127878. arXiv:2305.09014
  Abstract

  We consider constant mean curvature surfaces (invariant by a continuous group of isometries) lying at bounded distance from a horizontal geodesic on any homogeneous $3$-manifold $\mathbb{E}(\kappa,\tau)$ with isometry group of dimension $4$. These surfaces are called horizontal tubes. We show that they foliate $\mathbb{E}(\kappa,\tau)$ minus one or two horizontal geodesics provided that $(1-x_0^2)\kappa+4\tau^2\leq 0$, where $x_0\approx 0.833557$. We also describe precisely how horizontal and vertical geodesics get deformed by Daniel’s sister correspondence and conclude that the family of horizontal tubes is preserved by the correspondence. These tubes are topologically tori in $\mathbb{S}^2\times\mathbb{R}$ and Berger spheres, in which case we compute their conformal type and analyze numerically their isoperimetric profiles.

• Jesús Castro-Infantes, José M. Manzano, Genus one minimal k-noids and saddle towers in $\mathbb{H}^2\times \mathbb{R}$. J. Inst. Math. Jussieu, 22(5), 2023, 2155—2175.
  10.1017/S1474748021000591. arXiv:2001.07028
  Abstract

  For each $k \geq 3$, we construct a 1-parameter family of complete Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb{H}^2\times \mathbb{R}$ with genus $1$ and $k$ embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb{H}^2\times \mathbb{R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature $-4k\pi$. Finally, we provide examples of complete properly embedded minimal surfaces with infinitely many ends, each of them asymptotic to a vertical plane and with finite total curvature.

• Philipp Käse, Francisco Torralbo, Invariant constant mean curvature tubes in homogeneous spaces. J. Math. Anal. Appl., 556, 2025, 1—29.
  10.1016/j.jmaa.2025.130110. arXiv:2412.16070
  Abstract

  We study the global geometry of families of tubes of constant mean curvature invariant under screw-motions in homogeneous $\mathbb{E}(\kappa, \tau)$-spaces. In particular, we study embeddedness and prove a foliation result. Moreover, we study numerically the isoperimetric profile in the compact case.