A classical result of Nitsche \cite{Nit57} about the behaviour of the solutions to the Liouville equation $Δu=4 e^{2u}$ near isolated singularities is generalized to solutions of the Gaussian curvature equation
$Δu=- κ(z) e^{2u}$ where $κ$ is a negative Hölder continuous function. As an application a higher--order version of the Yau--Ahlfors--Schwarz lemma for complete conformal Riemannian metrics is obtained.
The precise asymptotic behaviour of the solutions to the twodimensional curvature equation $Δu=k(z) e^{2 u}$ with $e^{2 u} \in L^1$ for bounded nonnegative curvature functions $-k(z)$ near isolated singularities is obtained.
We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski $3$-space.
In this paper we derive a refined asymptotic expansion, near an isolated singularity, for conformally flat metrics with constant positive Q-curvature and positive scalar curvature. The condition that the metric has constant Q-curvature forces the conformal factor to satisfy a fourth order nonlinear partial differential equation with critical Sobolev growth, whose leading term is the bilaplacian. We model our results on a similar asymptotic expansion for conformally flat, constant scalar curvature metrics proven by Korevaar, Mazzeo, Pacard, and Schoen. Along the way we analyze the linearization…
Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.
We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves in the 2-sphere with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R^3, i.e. compact convex surfaces of constant positive curvature with a finite number of singularities, …
In this paper we study the short time existence problem for the (generalized) Lagrangian mean curvature flow in (almost) Calabi--Yau manifolds when the initial Lagrangian submanifold has isolated conical singularities modelled on stable special Lagrangian cones. Given a Lagrangian submanifold $F_0:L\rightarrow M$ in an almost Calabi--Yau manifold $M$ with isolated conical singularities at $x_1,...,x_n\in M$ modelled on stable special Lagrangian cones $C_1,...,C_n$ in $\mathbb{C}^m$, we show that for a short time there exist one-parameter families of points $x_1(t),... x_n(t)\in M$ and a one pa…
In this paper we prove that the set of metrics conformal to the standard metric on $\mathbb{S}^{n}\backslash\{p_{1},\cdots,p_{l}\}$ is locally compact in $C^{m,α}$ topology for any $m>0$, whenever the metrics have constant $σ_{k}$ curvature and the $k$-Dilational Pohozaev invariants have positive lower bound for $k<n/2$. Here the $k$-Dilational Pohozaev invariants come from the Kazdan-Warner type identity for the $σ_{k}$ curvature, which is derived by Viaclovsky \cite{Viac2000} and Han \cite{H1}. When $k=1$, Pollack \cite{Pollack} proved the compactness results for the complete metrics of cons…
In this paper we analyse the Euler implicit scheme for the volume preserving mean curvature flow. We prove the exponential convergence of the scheme to a finite union of disjoint balls with equal volume for any bounded initial set with finite perimeter.
We show a new example of blow-up behaviour for the prescribed $Q$-curvature equation in even dimension $6$ and higher, namely given a sequence $(V_k)\subset C^0(\mathbb{R}^{2n})$ suitably converging we construct {for $n\geq 3$} a sequence $(u_k)$ of radially symmetric solutions to the equation $${(-Δ)^n u_k=V_k e^{2n u_k} \quad \text{in }\mathbb{R}^{2n},}$$ with $u_k$ blowing up at the origin \emph{and} on a sphere. We also prove sharp blow-up estimates. This is in sharp contrast with the $4$-dimensional case studied by F. Robert (J. Diff. Eq. 2006).
We construct Kahler-Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity. Such metrics are complete near the singularity if the underlying space has complex dimension 2 or if the singularity is smoothable. In complex dimension 2, we show that any complete Kahler-Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to one of the model metrics constructed by Kobayashi and Nakamura arising from hyperbolic geometry.
We investigate the long-time behaviour of solutions to a nonlocal partial differential equation on smooth Riemannian manifolds of bounded sectional curvature. The equation models self-collective behaviour with intrinsic interactions that are modelled by an interaction potential. We consider attractive interaction potentials and establish sufficient conditions for a consensus state to form asymptotically. In addition, we quantify the approach to consensus, by deriving a convergence rate for the diameter of the solution's support. The analytical results are supported by numerical simulations for…
During his experiments W. D. Ristenpart made a very remarkable discovery. If two oppositely charged droplets of fluid are close enough, at first they attract each other and touch eventually. Surprisingly after that the droplets are repelled from each other, if the initial strength of the charges is high enough. Otherwise they coalesce and form a big drop, as one might expect. We present a theoretical model for this observation, using mean curvature flow. With the help of appropriate barriers for the flow we can predict the observed droplet behaviour. This shows that, contrary to general belief…
We show that the discrete approximate volume preserving mean curvature flow in the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set $E$ of the perimeter converges in the long time to a translate of $E$ exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided.
In 1996, Huisken-Yau showed that every three-dimensional Riemannian manifold can be uniquely foliated near infinity by stable closed CMC-surfaces if it is asymptotically equal to the (spatial) Schwarzschild solution and has positive mass. Their assumptions were later weakened by Metzger, Huang, Eichmair-Metzger and others. We further generalize these existence results in dimension three by proving that it is sufficient to assume asymptotic flatness and non-vanishing mass to conclude the existence and uniqueness of the CMC-foliation and explain why this seems to be the conceptually optimal resu…
We propose a new foliation of asymptotically Euclidean initial data sets by 2-spheres of constant spacetime mean curvature (STCMC). The leaves of the foliation have the STCMC-property regardless of the initial data set in which the foliation is constructed which asserts that there is a plethora of STCMC 2-spheres in a neighborhood of spatial infinity of any asymptotically flat spacetime. The STCMC-foliation can be understood as a covariant relativistic generalization of the CMC-foliation suggested by Huisken and Yau. We show that a unique STCMC-foliation exists near infinity of any asymptotica…
We study the asymptotic behaviour of the $p$-capacitary potential and of the weak Inverse Mean Curvature Flow of a bounded set along the ends of an Asymptotically Conical Riemannian manifolds with asymptotically nonnegative Ricci curvature.
We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We classify the geometrical cases we study into quasi-hyperbolic, quasi-Euclidean and critical cases, depending on the growth rate of the curvature at infinity. We prove sharp upper and lower bounds on the long-time behaviour of the solutions in terms of corresponding bounds on the curvature. In particular we obtain a sharp form of the smoothing effect on such ma…
We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are: first, to set the problem into a natural geometric framework, and second, to perform some kind of phase portrait study for this non-local ODE.
In the spirit of the paper "Large conformal metrics of prescribed Gauss curvature on surfaces of higher genus" by Borer-Galimberti-Struwe, where we dealt with the case of a closed Riemann surface $(M,g_0)$ of genus greater than one, here we study the behaviour of the conformal metrics $g_λ$ of prescribed Gauss curvature $K_{g_λ} = f_0 + λ$ on the torus, when the parameter $λ$ tends to one of the boundary points of the interval of existence of $g_λ$, and we characterize their "bubbling behaviour".
We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point $ζ\in\partialΩ\cup\{\infty\}$ of the quasilinear elliptic equations $$-\text{div}(|\nabla u|_A^{p-2}A\nabla u)+V|u|^{p-2}u =0\quad\text{in } Ω\setminus\{ζ\},$$ where $Ω$ is a domain in $\mathbb{R}^d$ ($d\geq 2$), and $A=(a_{ij})\in L_{\rm loc}^{\infty}(Ω;\mathbb{R}^{d\times d})$ is a symmetric and locally uniformly positive definite matrix. The potential $V$ lies in a certain local Morrey space (depending on $p$) and has a Fuchsian-type isolated singularity at $ζ$.
We consider the classical geometric problem of prescribing the scalar and boundary mean curvatures via conformal deformation of the metric on a $n-$dimensional compact Riemannian manifold. We deal with the case of negative scalar curvature and positive boundary mean curvature. It is known that if $n=3$ all the blow-up points are isolated and simple. In this work we prove that, for a linear perturbation, this is not true anymore in low dimensions $4\leq n\leq 7$. In particular, we construct a solution with a clustering blow-up boundary point (i.e. non-isolated), which is non-umbilic and is a lo…
In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a phaseshift $γ_l(t)$ that depends on the coordinate $l$ transverse to the primary direction of propagation. This direction is allowed to be general, but rational, generalizing earlier known results for the horizontal direction. We show that the behaviour of $γ$ can be asymptotically linked to the behaviour of a suitably discretized mean curvature flow. This allow…
$σ_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotic to a radial solution to the same equation on $\mathbb R^n \setminus \{0\}$. These results generalize earlier pioneering work in this direction on the classical Yamabe equa…
In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation $$ Δ^2 u = u^p~~~~~~~in ~ B_1 \backslash \{0\}$$ with an isolated singularity, where the punctured ball $B_1 \backslash \{0\} \subset \mathbb{R}^n$ with $n\geq 5$ and $\frac{n}{n-4} < p < \frac{n+4}{n-4}$. This equation is relevant for the $Q$-curvature problem in conformal geometry. We classify isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for $-Δu$. We also give a new method to prove removable singul…
We discuss compactness, blow-up and quantization phenomena for the prescribed $Q$-curvature equation $(-Δ)^m u_k=V_ke^{2mu_k}$ on open domains of $\R{2m}$. Under natural integral assumptions we show that when blow-up occurs, up to a subsequence $$\lim_{k\to \infty}\int_{Ω_0} V_ke^{2mu_k}dx=LΛ_1,$$ where $Ω_0\subset\subsetΩ$ is open and contains the blow-up points, $L\in\mathbb{N}$ and $Λ_1:=(2m-1)!\vol(S^{2m})$ is the total $Q$-curvature of the round sphere $S^{2m}$. Moreover, under suitable assumptions, the blow-up points are isolated. We do not assume that $V$ is positive.
On $(M,g)$ a compact riemannian $4-$manifold we consider the prescribed $Q-$curvature equation defined on $M$ with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on $\mathbb R^4$ and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an in…
We study the local properties of positive solutions of the equation $-Δu=u^p-m|\nabla u|^q$ in a punctured domain $Ω\setminus\{0\}$ of $\mathbb{R}^N$ or in a exterior domain $\mathbb{R}^N\setminus B_{r_0}$ in the range $\min\{p,q\}>1$ and $m>0$. We prove a series of a priori estimates depending $p$ and $q$, and of the sign of $q-\frac {2p}{p+1}$ and $q-p$. Using various techniques we obtain removability results for singular sets and we give a precise description of behaviour of solutions near an isolated singularity or at infinity in $\mathbb{R}^N$.
In this note we investigate the behaviour at finite-time singularities of the mean curvature flow of compact Riemannian submanifolds M^m_t\hookrightarrow (N^{m+n}, h). We show that they are characterized by the blow-up of a trace A = H \cdot II of the square of the second fundamental form.
We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this note is the introduction of a concept of mildly degenerate characteristic point for a smooth surface of the Heisenberg group, in a neighborhood of which the sub-Riemannian mean curvature is integrable (with respect to the perimeter measure induced by the Euclidean structure). As a consequence we partially answer to a question posed by Danielli-Garofalo-Nhieu in [Danielli D., Garofalo N., Nhieu D.M., Proc. Amer. Math. Soc.,…
We define a generalization of convex functions, which we call $δ$-convex functions, and show they must satisfy interior Hölder and $W^{1,p}$ estimates. As an application, we consider solutions of a certain class of fully nonlinear equations in conformal geometry with isolated singularities, in the case of non-negative Ricci curvature. We prove that such solutions either extend to a Hölder continuous function across the singularity, or else have the same singular behavior as the fundamental solution of the conformal Laplacian. We also obtain various removable singularity theorems for these equa…
We study the long-time behaviour of nonnegative solutions of the Porous Medium Equation posed on Cartan-Hadamard manifolds having very large negative curvature, more precisely when the sectional or Ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole. We find an unexpected separate-variable behaviour that reminds one of Dirichlet problems on bounded Euclidean domains. As a crucial step, we prove existence of solutions to a related sublinear elliptic problem, a result of independent interest. Uniqueness of solutions vanishing at infinity is …
