Related papers: Amoeba finite basis does not exist in general
The amoebas associated to algebraic varieties are certain concave regions in the Euclidean space whose shape reminds biological amoebas. This term was formally introduced to Mathematics in 1994 by Gelfand, Kapranov and Zelevinski. Some…
A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. Similarly, a non-archimedean coamoeba is the image of a subvariety of a torus over a non-archimedean field with complex residue field under…
In this article, we establish a relationship between geometric quantities of a hypersurface restricted to its boundary, and the geometric quantities of its boundary as a hypersurface of the boundary of the ball. As a first application, we…
We classify real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadrics ${Q^*}^{m} = SO^{o}_{2,m}/SO_mSO_2$, $m \geq 3$. We show that $m$ is even, say $m = 2k$, and any such hypersurface becomes an open part of a tube…
This is a case study of the algebraic boundary of convex hulls of varieties. We focus on surfaces in fourspace to showcase new geometric phenomena that neither curves nor hypersurfaces do. Our method is a detailed analysis of a general…
We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature…
We provide an infinite series of commutative finite-dimensional Gorenstein local algebras $A_n$ for $n \ge 2$. We give an elementary proof that the maximal ideal of every algebra $A_n$ possesses a one-dimensional subspace that is different…
We study the amoebae (in S. Favorov's sense) of finite systems of exponential sums with real spectra. For generic systems of this type, we give a more effective representation of the amoeba and we prove that the complementary set to the…
Coboundary expansion is a high dimensional generalization of the Cheeger constant to simplicial complexes. Originally, this notion was motivated by the fact that it implies topological expansion, but nowadays a significant part of the…
In this work we present a new family of properly embedded free boundary minimal hypersurfaces of revolution with circular boundaries in the horizon of the $n$-dimensional Schwarzschild space, $n\geq3$. In particular, we answer a question…
We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of…
We prove that a complete hyperbolic 3-manifold of finite volume does not admit a properly embedded noncompact surface of finite topology with constant mean curvature greater than or equal to 1.
Conformal fundamental forms populate a minimal generating set for low differential order invariants of conformal hypersurface embeddings. In this work we complete the characterization of conformal fundamental forms by proving the general…
In the first part of this paper, we establish some results around generalized Borel's Theorem. As an application, in the second part, we construct example of smooth surface of degree $d\geq 19$ in $\mathbb{CP}^3$ whose complements is…
In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a…
We use the specialization homomorphism for the birational automorphism group to study finite order birational automorphisms. For a family of varieties over a DVR, we prove that a birational automorphism of order coprime to the residue…
We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author.…
We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do…
An almost periodic function in finite-dimensional space extends to a holomorphic bounded function in a tube domain with a cone in the base if and only if the spectrum belongs to the conjugate cone. Also, an almost periodic function in…
We present a theory that produces several examples where the homotopy Lie algebra of a complex hyperplane arrangement is not finitely presented. We also present examples of hyperplane arrangements where the enveloping algebra of this Lie…