arXiv research

A quaternionic version of Picard's theorem limits how many values a slice regular function can avoid.

A new option pricing model handles non-constant risk aversion and transaction costs.

A new algorithm optimizes time-varying functions with non-constant evaluation times.

Develops discrete geometry for non-constant curvature surfaces.

It is well-known that non-constant holomorphic functions do not exist on a compact complex manifold. This statement is false for a supermanifold with a compact reduction. In this paper we study the question under what conditions non-constant holomorphic functions do not exist on a compact homogeneous complex supermanif…

In this paper we consider the Ricci curvature of a Ricci soliton. In particular, we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also i…

Conformally compact and complete smooth solutions to the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton using the first Pontrjagin form of the (-)-connection} on 6-dimensional non-Kaehler nilmanifold are presented. In the conformally compact case the dilaton is determined by t…

Study optimal portfolio strategies with time-varying discount rates.

The paper classifies hypersurfaces in $\mathbb{H}^2 imes\mathbb{H}^2$ with constant curvature.

In this paper we obtain a splitting theorem for the symmetric diffusion operator $Δ_φ=Δ-\left<\nablaφ,\nabla \right>$ and a non-constant $C^3$ function $f$ in a complete Riemannian manifold $M$, under the assumptions that the Ricci curvature associated with $Δ_φ$ satisfies ${\rm Ric}_φ(\nabla f,\nabla f)\ge 0$, that $|…

In this paper we define the Reidemeister torsion as a rational function on the geometric components of the character variety of a one-cusped hyperbolic manifold M. We study its poles and zeros, and we deduce sufficient conditions on the manifold M for this function being non-constant.

We show that if $p \colon M \to N$ is a normal Riemannian covering, with $N$ closed, and $M$ has exponential volume growth, then there are non-constant, positive harmonic functions on $M$. This was conjectured by Lyons and Sullivan in \cite{LS}.

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show that any Riemannian metric on $M$ admits infinitely many prime closed geodesics such that the energy functional $E:ΛM\to\m…

By using the Hawking Taub-NUT metric, this note gives an explicit construction of a 3-parameter family of Einstein Finsler metrics of non-constant flag curvature in terms of navigation representation.

Proves existence and uniqueness of Killing graphs with prescribed curvature.

We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.

We prove that for any open Riemann surface $M$ and any non constant harmonic function $h:M \to \mathbb{R},$ there exists a complete conformal minimal immersion $X:M \to \mathbb{R}^3$ whose third coordinate function coincides with $h.$ As a consequence, complete minimal surfaces with arbitrary conformal structure and wh…

Study on MHD equilibria on curved spaces without symmetries.

New smooth solutions of the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton based on the quaternionic Heisenberg group are constructed. We show that through appropriate contractions the solutions found in the $G_2$-heterotic case converge to the heterotic solutions on 6-dimensi…

To obtain groups with bounded harmonic functions (which are not hyperbolic), one of the most frequent way is to look at some semi-direct products (\eg lamplighter groups). The aim here is to show that many of these semi-direct products do not admit harmonic functions with gradient in $\ell^p$, for $p\in [1,\infty[$.

The paper extends Menelaus' and Ceva's theorems to translation triangles in various Thurston geometries.

We prove that there does not exist non-constant positive $f$-harmonic function on the complete gradient shrinking Ricci solitons. We also prove the $L^{p}(p\geq 1 \ {\rm or}\ 0<p\leq 1)$ Liouville theorems on the complete gradient shrinking Ricci solitons.

We prove that a Kleinian group $G$ acting upon $\mathbb{H}^{n}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic (torsion) elements are uniformly bounded. This is accomplished by developing a technique for mashing distinct fat triangulations while…

The paper splits manifolds using infinity harmonic functions with linear growth.

Paper proves March's criterion for transience on symmetric manifolds.

The study improves fundamental gap estimates for surfaces with non-constant positive curvature.

Paper classifies special slant surfaces with varying curvature.

We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold $M$ with only one end if $M$ has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constan…

We use a phase space analysis to give some classification results for rotational hypersurfaces in $\mathbb{R}^{n+1}$ whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in $\mathbb{S}^n$, we show that a Delaunay-type classification hold…

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…

Researchers found equations for special surfaces in curved spaces.

Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with $\…

In this paper, we study the dividend strategies for a shareholder with non-constant discount rate in a diffusion risk model. We assume that the dividends can only be paid at a bounded rate and restrict ourselves to the Markov strategies. This is a time inconsistent control problem. The extended HJB equation is given an…

Quadratic differentials on Riemann surfaces uniquely determine foliations.

Generic metrics on manifolds yield simple Steklov eigenvalues and Morse boundary functions.

Constant and symmetric price impact functions, most commonly used in agent-based market modelling, are shown to give rise to paradoxical and inconsistent outcomes in the simplest case of arbitrage exploitation when open-hold-close actions are considered. The solution of the paradox lies in the non-constant nature of re…

We give some concrete examples of moduli spaces of connections. Precisely, we explain how to explicitly construct the moduli spaces of rank 2 fuchsian systems and logarithmic connections on the Riemann sphere with 4 poles. The former ones are affine cubic surfaces; the latter ones are analytically isomorphic to affine …

We study in this work the existence of minimizing solutions to the critical-power type equation $\triangle_{\textbf{g}}u+h.u=f .u^{\frac{n+2}{n-2}}$ on a compact riemannian manifold in the limit case normally not solved by variational methods. For this purpose, we use a concept of "critical function" that was original…

Study Hamiltonian stationary Lagrangian surfaces in complex space forms.

The classical Brody's theorem asserts the equivalence between two notions of hyperbolicity for compact complex spaces, one named after Kobayashi and one expressed in terms of lack of non constant holomorphic entire functions (compactness is only used to prove the harder implication). We extend this theorem to Deligne-M…

We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent …

The aim of this paper is to classify three dimensional compact Riemannian manifolds $(M^{3},g)$ that admits a non-constant solution to the equation $-Δf g+Hess f-fRic=μRic+λg,$ for some special constants $(μ, λ)$, under assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we will…

In this paper we consider the complete biconservative surfaces in Euclidean space $\mathbb{R}^3$ and in the unit Euclidean sphere $\mathbb{S}^3$. Biconservative surfaces in 3-dimensional space forms are characterized by the fact that the gradient of their mean curvature function is an eigenvector of the shape operator,…

We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost function, and satisfies the PDE, boundary conditions, and additional regularizations. Th…

Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle $\mathscr{O}_{T_M^*}(1)$ is not pseudo-effective and $\mathscr{O}_{T_N^*}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex mani…

Salkowski \cite{salkow}, one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space $\e^3$. In this paper, we adapt definition of such curves to ti…

We prove that every non-constant quasiregular selfmap of the $n$-sphere $\mathbb{S}^{n}$ admits a harmonic extension to the hyperbolic space $\mathbb{H}^{n+1}$ for $n\ge 2$.

Solves supercritical dHYM on projective manifolds with specific conditions.