We employ curvature flows without global terms to seek strictly convex, spacelike solutions of a broad class of elliptic prescribed curvature equations in the simply connected Riemannian spaceforms and the Lorentzian de Sitter space, where the prescribed function may depend on the position and the normal vector. In particular, in the Euclidean space we solve a class of prescribed curvature measure problems, intermediate $L_p$-Aleksandrov and dual Minkowski problems as well as their counterparts, namely the $L_{p}$-Christoffel-Minkowski type problems. In some cases we do not impose any condition on the anisotropy except positivity, and in the remaining cases our condition resembles the constant rank theorem/convexity principle due to Caffarelli-Guan-Ma (Commun. Pure Appl. Math. 60 (2007), 1769--1791). Our approach does not rely on monotone entropy functionals and it is suitable to treat curvature problems that do not possess variational structures.
This paper defines a parabolic frequency for solutions of the heat equation along homothetically shrinking mean curvature flows and proves its monotonicity along such flows. As a corollary, frequency monotonicity provides a proof of backwards uniqueness. Additionally, for solutions of more general parabolic equations on mean curvature flow shrinkers, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.
In \cite{FGP}, Fei, Guo and Phong established a criteria for the long-time existence of their parabolic flow from $11$-dimensional supergravity, which involves Riemannian curvatures ${\rm Rm}(g(t))$ and 4-forms $F(t)$. In this paper, we obtain a new criteria for the long-time existence of the same flow, which involves only Ricci curvatures ${\rm Ric}(g(t))$, $F(t)$, but as well as $\nabla_{g(t)}F(t)$.
We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of curvature is a second order system of partial differential equations which we show decouples to highest order. We determine the conditions for this system to be parabolic and investigate the lower order terms. The zeroth order terms are shown to form a Hamiltonian system, which is therefore completely integrable. We find conditions on parabolic curvature flow…
We define a (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen-Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one-dimensional standing wave), the limit …
In this article we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral Brakke flows that connect Clifford tori to equatorial spheres, and study a family of such flows, in particular their symmetry properties. Our approach is based on the realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen, and Tonegawa, and it uses the classification of ancient gradient flows in sp…
We give a new proof of Brakke's partial regularity theorem up to C^{1,ς} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard's regularity theorem in the sense that the special time-independent case reduces to Allard's theorem.
If the Lorentzian norm on a maximal surface in the 3-dimensional Lorentz-Minkowski space $R_1^3$ is positive and proper, then the surface is relative parabolic. As a consequence, entire maximal graphs with a closed set of isolated singularities are relative parabolic. Furthermore, maximal and minimal graphs over closed starlike domains in $R_1^3$ and $R^3,$ respectively, are relative parabolic.
We study parabolic linear Weingarten surfaces in hyperbolic space $\rlopezh^3$. In particular, we classify two family of parabolic surfaces: surfaces with constant Gaussian curvature and surfaces that satisfy the relation $aκ_1+bκ_2=c$, where $κ_i$ are the principal curvatures, and $a,b$ and $c$ are constant.
We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in higher codimension is a nonlinear parabolic system where many of the methods used for hypersurfaces do not apply and uniqueness of cylindrical blowups remained a major open problem. Our results imply regularity of the singular set for the system.
In this article, we prove a sharp estimate for the solutions to parabolic equations on manifolds. Precisely, using symmetrization techniques and isoperimetric inequalities on Riemannian manifold, we obtain a Bandle's comparison on complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature respectively. Our results generalize Bandle's result [6] to Riemannian setting, and Talenti's comparison for elliptic equation on manifolds by Colladay-Langford-McDonald [12] and Chen-Li [9] to parabolic equations.
We derive a parabolic version of Omori-Yau maximum principle for a proper mean curvature flow when the ambient space has lower bound on $\ell$-sectional curvature. We apply this to show that the image of Gauss map is preserved under a proper mean curvature flow in euclidean spaces with uniform bounded second fundamental forms. This generalizes the result of Wang \cite{Wang} for compact immersions. We also prove a Omori-Yau maximum principle for properly immersed self-shrinkers, which improves a result in \cite{CJQ}.
A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kaehler metric of constant scalar curvature on the blow-up according to math.DG/0412405. We present a generalization of this construction to the case of parabolically polystable ruled surfaces. Thus we can produce numerous examples of Kaehler surfaces of constant scalar curvature with circle or toric symmetry.
In this note, we establish the dihedral rigidity phenomenon for a collection of parabolic polyhedrons enclosed by horospheres in hyperbolic manifolds, extending Gromov's comparison theory to metrics with negative scalar curvature lower bounds. Our result is a localization of the positive mass theorem for asymptotically hyperbolic manifolds. We also motivate and formulate some open questions concerning related rigidity phenomenon and convergence of metrics with scalar curvature lower bounds.
Let $(M^{n},g)$ be a complete Riemannian manifold. In this paper, we establish a space-time gradient estimates for positive solutions of nonlinear parabolic equations $$\partial_{t}u(x,t)=Δu(x,t)+a u(x,t)(\log u(x,t))^b + q(x,t)A(u(x,t)),$$ on geodesic balls $B(O,r)$ in $M$ with $0<r\leq r$ for $p>\frac{n}{2}$ when integral Ricci curvature $k(p,1)$ is small enough. By integrating the gradient estimates, we find the corresponding Harnack inequalities.
We consider a system of three surfaces, graphs over a bounded domain in ${\mathbb R}^2$, intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to $2π/3$.) For the corresponding two-dimensional parabolic free boundary problem we prove short-time existence of classical solutions (in parabolic Hölder spaces), for sufficiently regular initial data satisfying a compatibility condition.
We show that the image of a nonconstant conformal harmonic map $\mathbb C\to \mathbb R^3$, not necessarily proper and possibly with branch points, intersects every properly embedded nonflat minimal surface of bounded curvature in $\mathbb R^3$. The same holds if $\mathbb C$ is replaced by any open conformal surface without nonconstant bounded subharmonic functions.
A surface in hyperbolic space $\h^3$ invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of $\h^3$ that satisfy a linear Weingarten relation of the form $aκ_1+bκ_2=c$ or $aH+bK=c$, where $a,b,c\in \r$ and, as usual, $κ_i$ are the principal curvatures, $H$ is the mean curvature and $K$ is de Gaussian curvature. We classify all parabolic linear Weingarten surfaces in hyperbolic space.
We construct infinite families of regular normal Cartan geometries with nonvanishing curvature and essential automorphisms on closed manifolds for many higher rank parabolic model geometries. To do this, we use particular elements of the kernel of the Kostant Laplacian to construct homogeneous Cartan geometries of the desired type, giving a global realization of an elegant local construction due to Kruglikov and The, and then modify these homogeneous geometries to make their base manifolds compact. As a demonstration, we apply the construction to quaternionic contact structrures of mixed signa…
In 1993, Bartnik introduced a quasi-spherical construction of metrics of prescribed scalar curvature on 3-manifolds. Under quasi-spherical ansatz, the problem is converted into the initial value problem for a semi-linear parabolic equation of the lapse function. The original ansatz of Bartnik started with a background foliation with round metrics on the 2-sphere leaves. This has been generalized by several authors under various assumptions on the background foliation. In this article, we consider background foliations given by conformal round metrics, and by the Ricci flow on 2-spheres. We dis…
In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the $3$-dimensional Euclidean space or in the $3$-dimensional sphere is parabolic. In the case of ends of revolution in the hyperbolic $3$-dimensional space, we find sufficient conditions to attain parabolicity for complete ends of revolution using their relative position to the complete flat surfaces of revolution.
We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrodinger operator $-L=-(Δ+q)$ on a parabolic Riemannian manifold M, then u is either identically zero or it has no zeros on M, and the linear space of such functions is 1-dimensional. We obtain consequences for orientable, complete stable surfaces with constant mean curvature $H\in\mathbb{R}$ in homogeneous spaces $\mathbb{E}(κ,τ)$ with four dimensional isometry group. For instance, if M is an orientable, parabolic, complete immersed surface with constant mean curvature H in $\mathbb{H}^2\times\mathbb{R}$, then $…
We generalize the concept of affine locally symmetric spaces for parabolic geometries. We discuss mainly $|1|$--graded geometries and we show some restrictions on their curvature coming from the existence of symmetries. We use the theory of Weyl structures to discuss more interesting $|1|$--graded geometries which can carry a symmetry in a point with nonzero curvature. More concretely, we discuss the number of different symmetries which can exist at the point with nonzero curvature.
We study some potential theoretic properties of homothetic solitons $Σ^n$ of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in $\mathbb{R}^{n+m}$, we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons $Σ^n$ with $n>2$ are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the Mean Exit Time function for the Brow…
We review recent compactness and non-compactness results for the Yamabe equation. We also discuss the asymptotic behavior of the parabolic Yamabe flow.
In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and many nonlinear flows in the literature. We first show that the only compact self-expanders to any of these flows are round spheres. Secondly, we show that complete non-compact self-expanders to any of these flows with asymptotically cylindrical ends must be rotationally symmetric. Thirdly, we show that when such a flow is uniformly parabolic, there exist comp…
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…
We introduce a parabolic flow of almost Kahler structures, providing an approach to constructing canonical geometric structures on symplectic manifolds. We exhibit this flow as one of a family of parabolic flows of almost Hermitian structures, generalizing our previous work on parabolic flows of Hermitian metrics. We exhibit a long time existence obstruction for solutions to this flow by showing certain smoothing estimates for the curvature and torsion. We end with a discussion of the limiting objects as well as some open problems related to the symplectic curvature flow.
In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curvatures of the ambient symmetric spaces. Consequently, all of our solvmanifolds are Einstein, which provide a large number of new examples of noncompact homogeneous Einstein manifolds. We also show that our solvmanifolds are minimal, but not totally geodesic submanifolds of symmetr…
We consider a fully nonlinear parabolic equation with nonlinear Neumann type boundary condition, and show that the longtime existence and convergence of the flow. Finally we apply this study to the boundary value problem for minimal Lagrangian graphs.
We study surfaces of constant mean curvature which are invariant by oneparameter group of either rotational isometries or parabolic isometries, immersed into the homogeneous manifold PSL2(R,tau). Also, we give some applications.
In this talk I will discuss an example of the use of fully nonlinear parabolic flows to prove geometric results. I will emphasise the fact that there is a wide variety of geometric parabolic equations to choose from, and to get the best results it can be very important to choose the best flow. I will illustrate this in the setting of surfaces in a three-dimensional sphere. There are quite a few relevant results for surfaces in the sphere satisfying various kinds of curvature equations, including totally umbillic surfaces, minimal surfaces and constant mean curvature surfaces, and intrinsically…
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
We prove that the moduli of continuity of viscosity solutions to fully nonlinear parabolic partial differential equations are viscosity subsolutions of suitable parabolic equations of one space variable. As applications, we obtain sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data, via comparison with one-dimensional solutions. This work extends multiple results of Andrews and Clutterbuck for quasilinear equations to fully nonlinear equations.
I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate $G$-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quantities invariant up to a generalized change of variables). One family of invariants gives a geometric characterization for parabolic equation of Monge-Ampère type. A second family of invariants determines when a parabolic equation has a local choice of coordinates putting it in e…
In this paper, we prove Souplet-Zhang type gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with the compact boundary under the Dirichlet boundary condition when the Bakry-Emery Ricci tensor and the weighted mean curvature are both bounded below. As an application, we obtain a new Liouville type result for some space-time functions on such smooth metric measure spaces. These results generalize previous linear equations to a nonlinear case.
Let $(M,g(t))$, $0\le t\le T$, be a n-dimensional complete noncompact manifold, $n\ge 2$, with bounded curvatures and metric $g(t)$ evolving by the Ricci flow $\frac{\partial g_{ij}}{\partial t}=-2R_{ij}$. We will extend the result of L. Ma and Y. Yang and prove a local gradient estimate for positive solutions of the nonlinear parabolic equation $\frac{\1 u}{\1 t}=Δu-au\log u-qu$ where $a\in\R$ is a constant and $q$ is a smooth function on $M\times [0,T]$.
Second derivative pinching estimates are proved for a class of elliptic and parabolic equations, including motion of hypersurfaces by curvature functions such as quotients of elementary symmetric functions of curvature. The estimates imply convergence of convex hypersurfaces to spheres under these flows, improving earlier results of B. Chow and the author. The result is obtained via a detailed analysis of gradient terms in the equations satisfied by second derivatives.
I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial co…
We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.
We introduce certain energy functionals to the complex Monge-Ampere equation over a bounded domain with inhomogeneous boundary condition, and use these functionals to show the convergence of the solution to the parabolic Monge-Ampere equation.
We study the parabolic flow for generalized complex Monge-Ampère type equations on closed Hermitian manifolds. We derive {\em a priori} $C^\infty$ estimates for normalized solutions, and then prove the $C^\infty$ convergence.
We study vector fields generating a local flow by automorphisms of a parabolic geometry with higher order fixed points. We develop general tools extending the techniques of [1], [2], and [3]. We apply these tools to almost Grassmannian, almost quaternionic, and contact parabolic geometries, including CR structures, to obtain descriptions of the possible dynamics of such flows near the fixed point and strong restrictions on the curvature. In some cases, we can show vanishing of the curvature on a nonempty open set. Deriving consequences for a specific geometry entails evaluating purely algebrai…
We construct a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with lightlike axis and call these surfaces meridian surfaces of parabolic type. They are analogous to the meridian surfaces of elliptic or hyperbolic type. Using the invariants of these surfaces we give the complete classification of the meridian surfaces of parabolic type with constant Gauss curvature or constant mean curvature. We also classify the Chen meridian surfaces of parabolic type and the meridian surfaces of parabolic type with para…
In this paper are studied the simplest qualitative properties of asymptotic lines of a plane field in Euclidean space. These lines are the integral curves of the null directions of the normal curvature of the plane field, on the closure of the hyperbolic region, where the Gaussian curvature is negative. When the plane field is completely integrable, these curves coincides with the classical asymptotic lines on surfaces.
