Parabolic Manifolds Research Articles

A numerical study of the scattering in the He-Cu model with a Morse potential: Parabolic manifolds and exponentially small phenomena

We consider the classical approximation of a realistic model for the scattering of He atoms from Cu surfaces. For this problem, modeled by a two-degrees-of-freedom Hamiltonian system, the existence of chaos has been proven analytically very recently for sufficiently large values of the energy, if some quantity, known as the Stokes constant, is non-zero (Borondo et al., 2024). Taking two different and independent approaches, this paper provides numerical evidence that this is indeed the case. Both approaches provide the same value of the non-zero Stokes constant.

Polynomials on Regular Parabolic Manifolds

Polynomials on Regular Parabolic Manifolds

Poincaré constant on manifolds with ends

AbstractWe obtain optimal estimates of the Poincaré constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincaré constant is determined by the second largest end. The proof is based on the argument by Kusuoka–Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends.

On the Existence of Solutions of the Neumann Problem for the -Laplacian on Parabolic Manifolds with a Model End

A criterion for the existence of solutions of the second boundary value problem for the -Laplacian on Riemannian parabolic manifolds with a model end is obtained.

Birational properties of tangent to the identity germs without non‐degenerate singular directions

AbstractWe provide a family of isolated tangent to the identity germs which possess only degenerate characteristic directions, and for which the lift of to any modification (with suitable properties) has only degenerate characteristic directions. This is in sharp contrast with the situation in dimension 2, where any isolated tangent to the identity germ admits a modification where the lift of has a non‐degenerate characteristic direction. We compare this situation with the resolution of singularities of the infinitesimal generator of , showing that this phenomenon is not related to the non‐existence of complex separatrices for vector fields of Gomez‐Mont and Luengo. Finally, we describe the set of formal ‐invariant curves, and the associated parabolic manifolds, using the techniques recently developed by López‐Hernanz et al.

On the Existence of Solutions of the Neumann Problem for the $$p $$-Laplacian on Parabolic Manifolds with a Model End

On the Existence of Solutions of the Neumann Problem for the $$p $$-Laplacian on Parabolic Manifolds with a Model End

МАССИВНЫЕ МНОЖЕСТВА, ПОРОЖДЁННЫЕ ПОЛУЛИНЕЙНЫМИ ЭЛЛИПТИЧЕСКИМИ ОПЕРАТОРАМИ НА НЕКОМПАКТНЫХ РИМАНОВЫХ МНОГООБРАЗИЯХ

One of the origins of the topic of this study is the classification theory of non-compact Riemannian surfaces. It is well known that on parabolic surfaces, any superharmonic functions bounded from below is the identical constant. Hyperbolic surfaces contain nontrivial superharmonic functions. This distinct property of parabolic surfaces form the basis for the definitions of parabolic manifolds with dimensions greater than two. The classification theory of Riemannian manifolds is directly related to Liouville-type theorems which assert the triviality of bounded solutions of elliptic equations. High efficiency in this topic was shown by the capacitive technique developed in the works of Grigoryan, Losev, Mazepa, and others. In particular, estimates were obtained for the dimensions of bounded harmonic functions and solutions of the stationary Schrödinger equation on noncompact Riemannian manifolds in terms of massive sets. In this paper, we study the properties of massive sets generated by a semilinear elliptic operator. It was possible to prove that the property of massiveness is preserved under variations of the potential. The current work generalizes or strengthens the results of Mazepa. A necessary condition for the existence of nontrivial bounded solutions of a semilinear equation is also obtained.

Investigation of flow and heat transfer performance of the manifold microchannel with different manifold arrangements

The manifold microchannels (MMC) plays an essential role as an efficient method for heat dissipation in electronic devices. In this study, the flow and heat transfer characteristics of MMC with different manifold structures are investigated by numerical simulation. The results show the excellent effect of manifold optimization in alleviating the uneven fluid distribution between the microchannels and reducing the pressure loss of the MMC. It is observed that manifold optimization reduces the thermal resistance and temperature difference of the MMC. Finally, the comprehensive performance of different manifold structures is evaluated by the PEC value and their relationship among the pressure drop ratio, temperature difference ratio and thermal resistance ratio. And it is found that the parabolic-type manifold provides the best comprehensive performance. By analyzing the effect of conical manifold, parabolic manifold and elliptical manifold on the flow distribution, it was concluded that the uniform distribution of fluid in the MMC can be achieved by controlling the converging rate of the manifold. • The pressure drop of the manifold microchannel with tapered-type manifold is the smallest. • The optimized structures of manifold can improve uneven flow distribution among microchannels. • Parabolic-type manifold demonstrates the best comprehensive performance. • Uniform distribution of fluid can be achieved by controlling the arrangement of the manifold.

Polynomials on Regular Parabolic Manifolds

In this work, we consider the regular parabolic manifold X and polynomials on it. We prove some properties of regular parabolic manifolds and describe polynomials on complements of Weierstrass algebroidal sets.

Oscillatory motions and parabolic manifolds at infinity in the planar circular restricted three body problem

Consider the Restricted Planar Circular 3 Body Problem. If the trajectory of the body of zero mass is defined for all time, it can have the following four types of asymptotic motion when time tends to infinity forward or backward in time: bounded, parabolic (goes to infinity with asymptotic zero velocity), hyperbolic (goes to infinity with asymptotic positive velocity) or oscillatory (the position of the body is unbounded but goes back to a compact region of phase space for a sequence of arbitrarily large times). We consider realistic mass ratio for the Sun-Jupiter pair and Jacobi constant which allows the massless body to cross Jupiter's orbit. This is a non-perturbative regime. We prove the existence of all possible combinations of past and future final motions. In particular, we obtain the existence of oscillatory motions. All the constructed trajectories cross the orbit of Jupiter but avoid close encounters with it.The proof relies on analyzing the stable and unstable invariant manifolds of infinity and their intersections. We construct orbits shadowing these invariant manifolds by the method of correctly aligned windows. The proof is computer assisted.

Bernstein and Half-Space Properties for Minimal Graphs Under Ricci Lower Bounds

Abstract In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold $M$ with Ricci curvature bounded from below. This enables us to show that positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant and that complete, parabolic manifolds with Ricci curvature bounded from below have the half-space property. We avoid the need of sectional curvature bounds on $M$ by exploiting a form of the Ahlfors–Khas’minskii duality in nonlinear potential theory.

Second main theorem for meromorphic mappings intersecting moving targets on parabolic manifolds

In this paper, we establish a new second main theorem for meromorphic mappings from $M$ into $\mathbb{P}(V)$ intersecting moving targets $g_{j}:M\rightarrow\mathbb{P}(V^{\ast}),\ 1\leq j\leq q,$ where $M$ is a parabolic manifold and $V$ is a Hermitian vector space. As an application, we prove the algebraic dependence problem for meromorphic mappings with moving targets in general position.

Functional calculus of Laplace transform type on non-doubling parabolic manifolds with ends

Let $M$ be a non-doubling parabolic manifold with ends and $L$ a non-negative self-adjoint operator on $L^{2}(M)$ which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators $L = \Delta + V$ where $\Delta$ is the Laplace–Beltrami operator and $V$ is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of $L$ together with its time derivatives and then apply them to obtain the weak type $(1, 1)$ estimate of the functional calculus of Laplace transform type of $\sqrt{L}$ which is defined by $\mathfrak{M}(\sqrt{L}) f(x) := \int_{0}^{\infty} \bigl[\sqrt{L} e^{-t \sqrt{L}} f(x)\bigr] m(t) dt$ where $m(t)$ is a bounded function on $[0, \infty)$. In the setting of our study, both doubling condition of the measure on $M$ and the smoothness of the operators' kernels are missing. The purely imaginary power $L^{is}$, $s \in \mathbb{R}$, is a special case of our result and an example of weak type $(1, 1)$ estimates of a singular integral with non-smooth kernels on non-doubling spaces.

Boundedness of the Discrete Square Function on Nondoubling Parabolic Manifolds with Ends

Let $M$ be a nondoubling parabolic manifold with ends. In this study, we will investigate the $L^p$ boundedness of the discrete square function in terms of the Littlewood-Paley decomposition. It should be pointed out that, in our setting, the doubling condition of the underlying space and the regularity estimate of the kernel are missing.

CARTESIAN PRODUCT OF REGULAR PARABOLIC MANIFOLDS.

We consider the regular parabolic manifold X and polynomials on it. It is proved regularity of Cartesian product of regular parabolic manifolds and described polynomials.

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension $$n \ge 3$$ . For every bounded open subset $$\Omega \subset M$$ with smooth boundary, we prove that $$\begin{aligned} \int \limits _{\partial \Omega } \left| \frac{\mathrm{H}}{n-1}\right| ^{n-1} \!\!\!\!\!{\mathrm{d}}\sigma \,\,\ge \,\,{\mathrm{AVR}}(g)\,\big |\mathbb {S}^{n-1}\big |, \end{aligned}$$ where $${\mathrm{H}}$$ is the mean curvature of $$\partial \Omega $$ and $${\mathrm{AVR}}(g)$$ is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if $$(M{{\setminus }}\Omega , g)$$ is isometric to a truncated cone over $$\partial \Omega $$ . An optimal version of Huisken’s Isoperimetric Inequality for 3-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue’s non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.

L2-hard Lefschetz complete symplectic manifolds

For a complete symplectic manifold $M^{2n}$, we define the $L^{2}$-hard Lefschetz property on $M^{2n}$. We also prove that the complete symplectic manifold $M^{2n}$ satisfies $L^{2}$-hard Lefschetz property if and only if every class of $L^{2}$-harmonic forms contains a $L^{2}$ symplectic harmonic form. As an application, we get if $M^{2n}$ is a closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler characteristic satisfies the inequality $(-1)^{n}\chi(M^{2n})\geq0$.

Characteristic directions of two-dimensional biholomorphisms

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NONDOUBLING PARABOLIC MANIFOLDS WITH ENDS

AbstractLet $M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup ${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where $\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on $M$. Then, by combining the subordination formula with the previous result, we obtain the weak type $(1,1)$ and $L^{p}$ boundedness of the maximal function ${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on $L^{p}(M)$ for $1<p\leq \infty$ where $k$ is a nonnegative integer and $L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of $M$ and the smoothness of the operators’ kernels.

A Moser–Bernstein problem for Riemannian warped products

In this work we deal with an elliptic non-linear problem, which arises naturally from Riemannian geometry. This problem has classically been studied in the the Euclidean n-dimensional space and it is known as the Moser–Bernstein problem. Nevertheless we solve this type of problems in a wide family of Riemannian manifolds, constructed as Riemannian warped products. More precisely, we study the entire solutions to the minimal hypersurface equation in a Riemannian warped product $$M=P\times _h{\mathbb {R}}$$, where P is a complete Riemannian parabolic manifold and h a positive smooth function on P.