Related papers: A Non-Archimedean Second Main Theorem for Hypersur…
In this paper, we establish Liouville type results for semilinear subelliptic systems associated with the sub-Laplacian on the Heisenberg group $\mathbb{H}^{n}$ involving two different kinds of general nonlinearities. The main technique of…
We prove the hyperplane absolute winning property of weighted inhomogeneous badly approximable vectors in $\mathbb{R}^d$. This answers a question by Beresnevich--Nesharim--Yang and extends the main result of [Geometric and Functional…
Variational analysis provides the theoretical foundations and practical tools for constructing optimization algorithms without being restricted to smooth or convex problems. We survey the central concepts in the context of a concrete but…
The notion of an unexpected curve in the plane was introduced in 2018, and was quickly generalized in several directions in a flurry of mathematical activity by many authors. In this expository paper we first describe some of the main…
We proof existence theorems for the Dirichlet problem for hypersurfaces of constant special Lagrangian curvature in Hadamard manifolds. The first results are obtained using the continuity method and approximation and then refined using two…
For each pair $(Q_i,Q_j)$ of reference points and each real number $r$ there is a unique hyperplane $h \perp Q_iQ_j$ such that $d(P,Q_i)^2 - d(P,Q_j)^2 = r$ for points $P$ in $h$. Take $n$ reference points in $d$-space and for each pair…
In this paper, a Riemannian geometry of noncommutative super surfaces is developed which generalizes [4] to the super case. The notions of metric and connections on such noncommutative super surfaces are introduced and it is shown that the…
In this paper we further develop a Grassmannian technique used to prove results about very general hypersurfaces and present three applications. First, we provide a short proof of the Kobayashi Conjecture given previous results on the…
In this note we motivate the definition and use of Lie algebroids by revisiting the problem of reconstructing a hypersurface in Euclidean space from infinitesimal data.
We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of…
It was proved in Chen's paper \cite{chen} that every real hypersurface in the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$ \delta(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature and…
In this paper, we classify Euclidean umbilic-free hypersurfaces with semi-parallel Moebius second fundamental form and three distinct principal curvatures. This completes the classification of such hypersurfaces initiated by Hu, Xie and…
We show that Newton's method converges globally at a linear rate for objective functions whose Hessians are stable. This class of problems includes many functions which are not strongly convex, such as logistic regression. Our linear…
In this work we study analytic Levi-flat hypersurfaces in complex algebraic surfaces. First, we show that if this foliation admits chaotic dynamics (i.e. if it does not admit a transverse invariant measure), then the connected components of…
We prove an $l^p$ decoupling inequality for hypersurfaces with nonzero Gaussian curvature and use it to derive a corresponding $l^p$ decoupling for curves not contained in a hyperplane. This extends our earlier work from [2]
We explicitly determine the structure equations of 5-dimensional Levi 2-nondegenerate CR hypersurfaces, using our recently constructed canonical Cartan connection for this class of CR manifolds. We also give an outline of the basic…
We propose global surjectivity theorems of differentiable maps based on second order conditions. Using the homotopy continuation method, we demonstrate that, for a $C^2$ differentiable map from a Hilbert space to a finite-dimensional…
We show how to implement T-duality along non-abelian isometries in backgrounds with non-vanishing Ramond fields. When the dimension of the isometry group is odd (even) the duality swaps (preserves) the chirality of the theory. In certain…
In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called…
One version of the classical Lefschetz hyperplane theorem states that for $U \subset \mathbb P^n$ a smooth quasi-projective variety of dimension at least $2$, and $H \cap U$ a general hyperplane section, the resulting map on \'etale…