Minimal homogeneous submanifolds of complex hyperbolic spaces
We classify minimal extrinsically homogeneous submanifolds of complex hyperbolic spaces.
math.DG↗
arXiv subjects
Miguel Domínguez-Vázquez
Publications and source records attributed to Miguel Domínguez-Vázquez.
Positive $\mathrm{Ric}_2$ curvature on products of spheres and their quotients via intermediate fatness
We construct metrics of positive $2^{\rm nd}$ intermediate Ricci curvature, $\mathrm{Ric}_2>0$, on closed manifolds of dimensions 10, 11, 12, 13 and 14, including $\mathbb{S}^6\times\mathbb{S}^7$, $\mathbb{S}^7\times\mathbb{S}^7$ and all their simply connected isometric quotients. In particular, we obtain infinitely many examples in dimension 13. We also produce infinitely many non-simply connected spaces with $\mathrm{Ric}_2>0$ in dimensions 13 and 14, including $\mathbb{RP}^6\times \mathbb{RP}^7$ and $\mathbb{RP}^7\times \mathbb{RP}^7$, which cannot admit a metric of positive sectional curvature. The main new idea is a generalization of the concept of fatness which ensures the existence of $\mathrm{Ric}_2>0$ metrics on the total space of certain homogeneous bundles.
math.DG↗
Infinite families of manifolds of positive $k^{\rm th}$-intermediate Ricci curvature with $k$ small
Positive $k^{\rm th}$-intermediate Ricci curvature on a Riemannian $n$-manifold, to be denoted by $\mathrm{Ric}_k > 0$, is a condition that interpolates between positive sectional and positive Ricci curvature (when $k =1$ and $k=n-1$ respectively). In this work, we produce many examples of manifolds of $\mathrm{Ric}_k > 0$ with $k$ small by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension $n\geq 7$ congruent to $3\,\mathrm{mod}\ 4$ supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of $\mathrm{Ric}_k > 0$ for some $k 0$ with $k\leq n/2$, but do not admit metrics of positive sectional curvature.
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Overdetermined boundary problems with nonconstant Dirichlet and Neumann data
In this paper we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$, where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. Our main result is that, nevertheless, under minor technical hypotheses nontrivial solutions to the overdetermined boundary problem always exist.
math.AP↗
Piecewise smooth stationary Euler flows with compact support via overdetermined boundary problems
We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin-La-Vicol; in particular, these solutions are not localizable. A key step in the proof is the construction of solutions to an overdetermined elliptic boundary value problem where one prescribes both Dirichlet and (nonconstant) Neumann data.
math.AP↗
Isoparametric surfaces in $\mathbb{E}(κ,τ)$-spaces
We provide an explicit classification of the following four families of surfaces in any homogeneous 3-manifold with 4-dimensional isometry group: isoparametric surfaces, surfaces with constant principal curvatures, homogeneous surfaces, and surfaces with constant mean curvature and vanishing Abresch-Rosenberg differential.
math.DG↗