In this paper we study $k$-equivariant wave maps from the hyperbolic plane into the $2$-sphere as well as the energy critical equivariant $SU(2)$ Yang-Mills problem on $4$-dimensional hyperbolic space. The latter problem bears many similarities to a $2$-equivariant wave map into a surface of revolution. As in the case of $1$-equivariant wave maps considered in~\cite{LOS1}, both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schrödinger operator obtained by linearizing about such a harmonic map admits a simple positive eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models.
We show that the third eigenvalue of the Neumann Laplacian in hyperbolic space is maximal for the disjoint union of two geodesic balls, among domains of given volume. This extends a recent result by Bucur and Henrot in Euclidean space, while providing a new proof of a key step in their argument
In the paper, we investigate a local boundary value problem with transmitting condition of the integral form for mixed parabolic-hyperbolic equation with non-characteristic line of type changing. Two theorems on strong solvability and the existence of eigenvalues of the considered problem have been proved.
Optimisation of the lowest Robin eigenvalue in exterior domains of the hyperbolic plane
Università degli studi di Napoli Federico II（那不勒斯费德里科二世大学）；Czech Technical University in Prague（布拉格捷克理工大学）；Nuclear Physics Institute, Czech Academy of Sciences（捷克科学院核物理研究所）
In this paper, we prove the uniform estimates for the resolvent $(Δ- α)^{-1}$ as a map from $L^q$ to $L^{q'}$ on real hyperbolic space $\mathbb{H}^n$ where $α\in \mathbb{C}\setminus [(n - 1)^2/4, \infty)$ and $2n/(n + 2) \leq q < 2$. In contrast with analogous results on Euclidean space $\mathbb{R}^n$, the exponent $q$ here can be arbitrarily close to $2$. This striking improvement is due to two non-Euclidean features of hyperbolic space: the Kunze-Stein phenomenon and the exponential decay of the spectral measure. In addition, we apply this result to the study of eigenvalue bounds of the Schr…
We consider non-scattering energies and transmission eigenvalues of compactly supported potentials in the hyperbolic spaces $\mathbb H^n$. We prove that in $\mathbb H^2$ a corner bounded by two hyperbolic lines intersecting at an angle smaller than $180^\circ$ always scatters, and that one of the lines may be replaced by a horocycle. In higher dimensions, we obtain similar results for corners bounded by hyperbolic hyperplanes intersecting each other pairwise orthogonally, and that one of the hyperplanes may be replaced by a horosphere. The corner scattering results are contrasted by proving di…
Let $X$ be a compact connected hyperbolic surface, that is, a closed connected orientable smooth surface with a Riemannian metric of constant curvature -1. For each $n\in\mathbf{N}$, let $X_{n}$ be a random degree-$n$ cover of $X$ sampled uniformly from all degree-$n$ Riemannian covering spaces of $X$. An eigenvalue of $X$ or $X_{n}$ is an eigenvalue of the associated Laplacian operator $Δ_{X}$ or $Δ_{X_{n}}$. We say that an eigenvalue of $X_n$ is new if it occurs with greater multiplicity than in $X$. We prove that for any $\varepsilon>0$, with probability tending to 1 as $n\to\infty$, there …
We prove that if $X$ is a finite area non-compact hyperbolic surface, then for any $ε>0$, with probability tending to one as $n\to\infty$, a uniformly random degree $n$ Riemannian cover of $X$ has no eigenvalues of the Laplacian in $[0,\frac{1}{4}-ε)$ other than those of $X$, and with the same multiplicities. As a result, using a compactification procedure due to Buser, Burger, and Dodziuk, we settle in the affirmative the question of whether there exist a sequence of closed hyperbolic surfaces with genera tending to infinity and first non-zero eigenvalue of the Laplacian tending to $\frac{1}{…
In this paper we introduce and study a new kind of hyperbolic geometric flows --dissipative hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear wave equations with dissipative terms. Some interesting exact solutions are given, in particular, a new concept-- hyperbolic Ricci soliton is introduced and some of its geometric properties are described. We also establish the short-time existence and uniqueness theorem for the dissipative hyperbolic geometric flow, and prove the nonlinear stability of the flow defined on the Euclidean space of dimension larger than 2. Wa…
In this paper we investigate the life-span of classical solutions to the hyperbolic geometric flow in two space variables with slow decay initial data. By establishing some new estimates on the solutions of linear wave equations in two space variables, we give a lower bound of the life-span of classical solutions to the hyperbolic geometric flow with asymptotic flat initial Riemann surfaces.
Let M = R n or possibly a Riemannian, non compact manifold. We consider semi-excited resonances for a h-differential operator H(x, hD x ; h) on L 2 (M) induced by a non-degenerate periodic orbit $γ$ 0 of semi-hyperbolic type, which is contained in the non critical energy surface {H 0 = 0}. By semi-hyperbolic, we mean that the linearized Poincar{é} map dP 0 associated with $γ$ 0 has at least one eigenvalue of modulus greater (or less) than 1, and one eigenvalue of modulus equal to 1, and by non-degenerate that 1 is not an eigenvalue, which implies a family $γ$(E) with the same properties. It is…
In this paper we study the well-posedness of weakly hyperbolic systems with time dependent coefficients. We assume that the eigenvalues are low regular, in the sense that they are Hölder with respect to $t$. In the past these kind of systems have been investigated by Yuzawa \cite{Yu:05} and Kajitani \cite{KY:06} by employing semigroup techniques (Tanabe-Sobolevski method). Here, under a certain uniform property of the eigenvalues, we improve the Gevrey well-posedness result of \cite{Yu:05} and we obtain well-posedness in spaces of ultradistributions as well. Our main idea is a reduction of the…
We prove the SBV regularity of the characteristic speed of the scalar hyperbolic conservation law and SBV-like regularity of the eigenvalue functions of the Jacobian matrix of flux function for general systems of conservation laws. More precisely, for the equation u_t + f(u)_x = 0, \quad u : \R^+ \times \R \to Ω\subset \R^N, we only assume the flux $f$ is $C^2$ function in the scalar case (N=1) and Jacobian matrix $Df$ has distinct real eigenvalues in the system case $(N\geq 2)$. Using the modification of the main decay estimate in Lau and localization method applied in \cite{R}, we show that …
In this paper, we study the stabilization problem for a hyperbolic type Stokes system posed on a bounded domain. We show that when the damping effects are restricted to a subdomain satisfying the geometrical control condition the system decays exponentially. The result is a consequence of a new quasi-mode estimate for the Stokes system.
In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains of a Euclidean space or a hyperbolic space which strengthens the well-known Szegö-Weinberger inequality and supports strongly an important conjecture of Ashbaugh-Benguria.
In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ${\mathbb{R}}^{2}$ in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric $g_{ij}$ keeps uniformly bounded for all time. If the initial velocity tensor does not satisfy t…
In this work we want to explore the relationship between certain eigenvalue condition for the symbols of first order partial differential operators describing evolution processes and the linear and nonlinear stability of their stationary solutions.
We present a new approach to analyze the validation of weakly nonlinear geometric optics for entropy solutions of nonlinear hyperbolic systems of conservation laws whose eigenvalues are allowed to have constant multiplicity and corresponding characteristic fields to be linearly degenerate. The approach is based on our careful construction of more accurate auxiliary approximation to weakly nonlinear geometric optics, the properties of wave front-tracking approximate solutions, the behavior of solutions to the approximate asymptotic equations, and the standard semigroup estimates. To illustrate …
In 1996, Huisken-Yau proved that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed surfaces of constant mean curvature (CMC) if it is asymptotically equal to the (spatial) Schwarzschild solution. Using their method, Rigger proved the same theorem for Riemannian manifolds being asymptotically equal to the (spatial) (Schwarzschild-)Anti-de Sitter solution. This was generalized to asymptotically hyperbolic manifolds by Neves-Tian, Chodosh, and the author at a later stage. In this work, we prove the reverse implication as the author already did in …
