Blow-up analysis of Large conformal metrics with prescribed Gaussian and geodesic curvatures
Rayssa Caju, Tiarlos Cruz, Almir Silva Santos
备注：
Final version. 38 pages. References updated and some typos fixed. Improved version thanks to the anonymous referee. To appear in Calc. Var. Partial Differential Equations
Consider a compact Riemannian surface $(M,g)$ with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions $f$ in $M$ and $h$ in $\partial M$ with $\max f= \max h= 0$, under a suitable condition on the maximum points of $f$ and $h$, we prove that for sufficiently small positive constants $λ$ and $μ$, there exist at least two distinct conformal metrics $g_{λ,μ}=e^{2u_{μ,λ}}g$ and $g^{λ,μ}=e^{2u^{μ,λ}}g$ with prescribed sign-changing Gaussian and geodesic curvature equal to $f + μ$ and $h + λ,$ respectively. Additionally, we employ the method used in Borer et al. (2015) to study the blowing up behavior of the large solution $u^{μ,λ}$ when $μ\downarrow 0$ and $λ\downarrow 0$. Finally, we derive a new Liouville-type result for the half-space, eliminating one of the potential blow-up profiles.
We construct minimal laminations with prescribed singularities on a line segment using perturbation techniques and PDE methods. In addition to the singular set, the rate of curvature blowup is also prescribable in our construction, and we show that all curvature blowup rates between quadratic and quartic arise. Our result generalizes an earlier result of the author and of Hoffman and White.
In this paper we prove that all initially-smooth solutions of the Euler-Weil-Petersson equation, which describes geodesics on the universal Teichmüller space under the Weil-Petersson metric, will remain smooth for all time. This extends the work of Escher-Kolev for strong Riemannian metrics to the borderline case of $H^{3/2}$ metrics. In addition we show that all initially-smooth solutions of the Wunsch equation, which describes geodesics on the universal Teichmüller curve under the Velling-Kirillov metric, must blow up in finite time due to wave breaking, extending work of Castro-Córdoba and …
We construct a sequence of compact embedded minimal disks in a ball in Euclidean 3-space, whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the x_3-axis. This extends a result of Colding and Minicozzi, who constructed a sequence for which the curvature blows up only at the center of the ball, and is a partial affirmative answer to the larger question of the existence of a sequence for which the curvature blows up precisely on a prescribed closed set on the x_3-axis.
This paper is concerned with the existence of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures. Being more specific, given nonnegative smooth functions $K: \overline{\mathbb{D}} \to \mathbb{R}$ and $h: \partial \mathbb{D} \to \mathbb{R}$, we consider the problem of finding a conformal metric realizing $K$ and $h$ as Gaussian and geodesic curvatures, respectively. This is the natural analogue of the classical Nirenberg problem posed on the disk. As we shall see, both curvatures play a role in the existence of solutions. Indeed we are able to give existence results …
For any prescribed closed subset of a line segment in Euclidean 3-space, we construct a sequence of minimal disks that are properly embedded in an open solid cylinder around the segment and that have curvatures blowing up precisely at the points of the closed set.
This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents of Carnot geodesics are geodesics in some groups of lower nilpotency step. Namely, every blowup curve of every geodesic in every Carnot group is still a geodesic in the group modulo its last layer. Then as a consequence we get that in every sub-Riemannian manifold any $s$ times …
We construct all Finsler metrics on the two-sphere for which geodesics are circles and show that any (reversible) path geometry on a two-dimensional manifold is locally the system of geodesics of a Finsler metric.
Motivated by the results of B. Berndtsson, in this memoir we use the new estimates developed by W. He to extend a theorem of the second author on the existence of weak $C^{1,1}$ geodesics between two smooth non-degenerate Kähler potentials to the case where the metrics on the end points may have singularities on some analytic set and may be positive semi-definite.
This short note has been written as an Oberwolfach report for the workshop "Differentialgeometrie im Grossen". We discuss properties of metric spaces that at almost all points admit a tangent metric space. We explain why, under some mild assumptions, the tangents are almost surely subFinsler Carnot groups. We mention several situations when the tangents are Euclidean spaces, but the initial metric spaces are quite pathological.
In this paper we prescribe a fourth order conformal invariant on the standard $n-$sphere, with $n\geq5$, and study the related fourth order elliptic equation. We first find some existence results in the perturbative case. After some blow up analysis we build a homotopy to pass from the perturbative case to the non-perturbative one under some flatness condition. Finally we state some existence results under the assumption of symmetry.
We obtain simple characterizations of the connected components of the space of closed curves on the 2-sphere whose geodesic curvatures are constrained to lie in an open interval $(κ_1,κ_2)$, in terms of $κ_1$ and $κ_2$. Many results concerning the topology of these spaces are established. In particular, we determine the homeomorphism class of some of these components.
We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with at least two boundary components. This seems to be the first result in this setting. Moreover, we allow to have conical singularities with both positive and negative orders, that is cone angles both less and grater than $2π$.
We apply a local systolic-diastolic inequality for contact forms and odd-symplectic forms on three-manifolds to bound the magnetic length of closed curves with prescribed geodesic curvature (also known as magnetic geodesics) on an oriented closed surface. Our results hold when the prescribed curvature is either close to a Zoll one or large enough.
We prove the existence of Alexandrov embedded closed magnetic geodesics on closed hyperbolic surfaces. Closed magnetic geodesics correspond to closed curves with prescribed geodesic curvature.
We study singularity formation in spherically symmetric solutions of the charge-one and charge-two sector of the (2+1)-dimensional S^2 sigma-model and the (4+1)-dimensional Yang-Mills model, near the adiabatic limit. These equations are non-integrable, and so studies are performed numerically on rotationally symmetric solutions using an iterative finite differencing scheme that is numerically stable. We evaluate the accuracy of predictions made with the geodesic approximation. We find that the geodesic approximation is extremely accurate for the charge-two sigma-model and the Yang-Mills model,…
We give existence results for simple closed curves with prescribed geodesic curvature on $S^{2}$, which correspond to periodic orbits of a charge in a magnetic field.
We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.
We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time $T$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter $γ>1/2$, the…
In this paper, we study the prescribed $Q$-curvature flow equation on a arbitrary even dimensional closed Riemannian manifold $(M,g)$, which was introduced by S. Brendle in \cite{B2003}, where he proved the flow exists for long time and converges at infinity if the GJMS operator is weakly positive with trivial kernel and $\int_M Qdμ< (n-1)!\Vol\left( S^n \right) $. In this paper we study the critical case that $\int_M Qdμ= (n-1)!\Vol\left( S^n \right)$, we will prove the convergence of the flow under some geometric hypothesis. In particular, this gives a new proof of Li-Li-Liu's existence resu…
We construct a black hole initial data for the Einstein equations with prescribed scalar curvature, or more precisely a piece of initial data contained inside the black hole. The constraints translate into a parabolic equation, with radius as "time" variable, on a metric component u that undergoes blow up. The metric itself is regular up to and including the surface at blow up radius, which is a minimal surface. By applying equivariant bifurcation theory on a self similarly rescaled equation, we show the existence of blow up profiles that are not O(3) symmetric - or anisotropic - although the …
Given a constant $k>1$ and a real valued function $K$ on the hyperbolic plane $\mathbb H^2$, we study the problem of finding, for any $ε\approx 0$, a closed and embedded curve $u^ε$ in $\mathbb H^2$ having geodesic curvature $k+εK(u^ε)$ at each point.
We study "flat knot types" of geodesics on compact surfaces M^2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M^2. We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial.
We prove that on a closed surface, for any $c>0$, our min-max theory for prescribing mean curvature produces a solution given by a curve of constant geodesic curvature $c$ which is almost embedded, except for finitely many points, at which the solution is a stationary junction with integer density. Moreover, each smooth segment has multiplicity one. The key is a classification of blowups which is new even for $c=0$.
We prove a sharp area estimate for minimal submanifolds that pass through a prescribed point in a geodesic ball in hyperbolic space, in any dimension and codimension. In certain cases, we also prove the corresponding estimate in the sphere. Our estimates are analogous to those of Brendle and Hung in the Euclidean setting.
We consider surfaces in ${\mathbb R}^3$ of type ${\mathbb S}^2$ which minimize the Willmore functional with prescribed isoperimetric ratio. The existence of smooth minimizers was proved by Schygulla (Archive Rational Mechanics and Analysis, 2012). In the singular limit when the isoperimetric ratio converges to zero, he showed convergence to a double round sphere in the sense of varifolds. Here we give a full blowup analysis of this limit, showing that the two spheres are connected by a catenoidal neck. Besides its geometric interest, the problem was studied as a simplified model in the theory …
Initiated by the work of Uhlenbeck in late 1970s, we study questions about the existence, multiplicity and asymptotic behavior for minimal immersions of closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions. In particular, we determine when exactly the solution is unique and when multiple solutions appear. Moreover, we analyze in detail the asymptotic behavior of the solutions when (and how) blowing up might occur. Interestingly the blow-up analysis exhibit …
Let $C_{k_1}^{k_2}$ denote the set of all closed curves of class $C^r$ on the sphere $S^2$ whose geodesic curvatures are restricted to lie in $(k_1,k_2)$, furnished with the $C^r$ topology (for some $r >= 2$ and possibly infinite $k_1 < k_2$). In 1970, J. Little proved that the space $C_0^{+\infty}$ of closed curves having positive geodesic curvature has three connected components. Let $r_i = arccot k_i$ (i = 1, 2). We show that $C_{k_1}^{k_2}$ has n connected components $C_1, ..., C_n$, where n is the greatest integer smaller than or equal to $π/(r_1-r_2) + 1$, and $C_j$ contains circles trav…
In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with dimensions at least $ 3 $. The results are classified by the sign of the first eigenvalue of the conformal Laplacian. This leads to a "Trichotomy Theorem" in terms of both scalar and mean curvature functions, which is a full extension of the "Trichotomy Theorem" given by Kazdan and Warner. We also discuss prescribing Gauss and geodesic curvature problems on compact R…
In this paper we prove classification results to elliptic fully nonlinear conformal equations on certain subdomains of the sphere with prescribed constant mean curvature on its boundary. Such subdomains are the hemisphere (or a geodesic ball on $\mathbb{S}^n$) of dimension $n\geq 2$ with prescribed constant mean curvature on its boundary, and annular domains with minimal boundary. Our results extend the classifications of Escobar in \cite{E0} when $n\geq 3$, and Hang-Wang in \cite{HaWa} and Jimenez in \cite{J} when $n=2$.
In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the $(-1,-1,-1)$ type generalized circle packing metric introduced by Guo-Luo \cite{GL2}. To find hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths, we introduce combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings on ideally triangulated surfaces with boundary. Then we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows, which provide e…
We prove that a critical metric of the volume functional on a $4$-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $\mathbb{R}^{4}$, $\mathbb{H}^{4}$ or $\mathbb{S}^{4}.$ Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics.
We study the one-parameter family of twisted Kahler Taub-NUT metrics (discovered by Donaldson), along with two exceptional Taub-NUT-like instantons, and understand them to the extend that should be sufficient for blow-up and gluing arguments. In particular we parametrize their geodesics from the origin, determine curvature fall-off rates, volume growth rates for metric balls, and find blow-down limits.
We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature $H$ in warped product manifolds $M\times_\varrho \mathbb{R}$. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on $H$ and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curva…
We investigate the prescribed Q-curvature flow for GJMS operators with non-trivial kernel on compact manifolds of even dimension. When the total Q-curvature is negative, we identify a conformally invariant condition on the nodal domains of functions in the kernel of the GJMS operator, allowing us to prove the global existence and the convergence of the flow to a metric which is conformal to the initial one, and having a prescribed Q-curvature. If the total Q-curvature is positive, we show that the flow blows up in finite time.
We discover new monotonicity formulae for minimal submanifolds in space forms, which imply the sharp area bound for minimal submanifolds through a prescribed point in a geodesic ball. These monotonicity formulae involve an energy-like integral over sets which are, in general, not geodesic balls. In the Euclidean case, these sets reduce to the moving-centre balls introduced by the second author in [Zhu18].
We study that the $n-$graphs defining by smooth map $f:\Om\subset \ir{n}\to \ir{m}, m\ge 2,$ in $\ir{m+n}$ of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le\f{C}{R^2}$$ under the dimension limitations and the Gauss image restrictions. If there is no dimension limitation we obtain $$\sup_{D_R(x)}|B|^2\le C R^{-a}\sup_{D_{2R}(x)}(2-\De_f)^{-(\f{3}{2}+\f{1}{s})}, \qquad s=\min(m, n)$$ with $a<1$ under the condition $$\De_f=[\text{det}(\de_{ij}+\sum_\a\pd{f^\a}{x^i}\pd{f^\a}{x^j})]^{\f{1}{2}}<2.$$ If the image under the Gauss m…
The \textit{parabolic scalar curvature equation} is a reaction-diffusion type equation on an $(n-1)$-manifold $Σ$, the time variable of which shall be denoted by $r$. Given a function $R$ on $[r_0,r_1)\timesΣ$ and a family of metrics $γ(r)$ on $Σ$, when the coefficients of this equation are appropriately defined in terms of $γ$ and $R$, positive solutions give metrics of prescribed scalar curvature $R$ on $[r_0,r_1)\timesΣ$ in the form \[ g=u^2dr^2+r^2γ.\] If the area element of $r^2γ$ is expanding for increasing $r$, then the equation is parabolic, and the basic existence problem is to take p…
Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling…
