We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinking equatorial spheres. In the mean curvature flow case in S^3, we classify ancient flows with more relaxed area bounds: they are steady or shrinking equators or Clifford tori. In the embedded curve shortening case in S^2, we completely classify ancient flows of bounded length: they are steady or shrinking circles.
We show the existence of non-homothetic ancient flows by powers of curvature embedded in $\mathbb{R}^2$ whose entropy is finite. We determine the Morse indices and kernels of the linearized operator of shrinkers to the flows and construct ancient flows by using unstable eigenfunctions of the linearized operator.
In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of barycenters, we are able to establish a direct and geometrically clear degree counting formula for our problem. We remark that this formula has been proven with complete different methods by Lin Wei and Wang by using blow-up type estimates.
Given (M, g0) we consider the problem -ε^2Delta_{g0+h}u + u = (u+)^{p-1} with (ε, h) \in (0, ε0) \times Bρ. Here Bρ is a ball centered at 0 with radius ρ in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincaré polynomial of M, we give an estimate on the number of nonconstant solutions with low energy for (ε, h) belonging to a residual subset of (0, ε0) \times Bρ, for (ε0, ρ) small enough.
We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems, where the bifurcation parameter is introduced by shrinking the domain to a point. This extends recent results of the authors for scalar equations.
An explicit representation formula for all positive ancient solutions of the heat equation in the Euclidean case is found. In the Riemannian case with nonnegative Ricci curvature, a similar but less explicit formula is also found. Here it is proven that any positive ancient solution is the standard Laplace transform of positive solutions of the family of elliptic operator $Δ- s$ with $s>0$. Further relaxation of the curvature assumption is also possible. It is also shown that the linear space of ancient solutions of polynomial growth has finite dimension and these solutions are polynomials in …
We lay the foundations of a Morse homology on the space of connections on a principal $G$-bundle over a compact manifold $Y$, based on a newly defined gauge-invariant functional $\mathcal J$. While the critical points of $\mathcal J$ correspond to Yang-Mills connections on $P$, its $L^2$-gradient gives rise to a novel system of elliptic equations. This contrasts previous approaches to a study of the Yang-Mills functional via a parabolic gradient flow. We carry out the complete analytical details of our program in the case of a compact two-dimensional base manifold $Y$. We furthermore discuss i…
We give a short proof of the Morse index theorem for geodesics in semi-Riemannian manifolds by using K-theory. This makes the Morse index theorem reminiscent of the Atiyah-Singer index theorem for families of selfadjoint elliptic operators.
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…
In this paper, we study elliptic gradient estimates for a nonlinear $f$-heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton's and Souplet-Zhang's gradient estimates for positive solutions to the nonlinear $f$-heat equation only assuming the Bakry-Émery Ricci tensor is bounded below. As applications, we prove parabolic Liouville properties for some kind of ancient solutions to the nonlinear $f$-heat equation. Some special cases are also discussed.
Let $(M^N, g, e^{-f}dv)$ be a complete smooth metric measure space with $\infty$-Bakry-Émery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation \begin{align*} \displaystyle \Big(Δ_f - \frac{\partial}{\partial t}\Big) u(x,t) +q(x,t)u^α(x,t) = 0, \end{align*} where $(x,t) \in M^N \times (-\infty, \infty)$ and $α$ is an arbitrary constant. As Applications we prove a Liouville-type theorem for positive ancient solutions and Harnack-type inequalities for positive bounded solutions.
We prove a theorem about elliptic operators with symmetric potential functions, defined on a function space over a closed loop. The result is similar to a known result for a function space on an interval with Dirichlet boundary conditions. These theorems provide accurate numerical methods for finding the spectra of those operators over either type of function space. As an application, we numerically compute the Morse index of constant mean curvature tori of revolution in the unit 3-sphere $\mathbb{S}^3$, confirming that every such torus has Morse index at least five, and showing that other kno…
Given a compact four-dimensional Riemannian manifold $(M, g)$ with boundary, we study the problem of existence of Riemannian metrics on $M$ conformal to $g$ with prescribed $Q$-curvature in the interior $\mathring{M}$ of $M$, and zero $T$-curvature and mean curvature on the boundary $\partial M$ of $M$. This geometric problem is equivalent to solving a fourth-order elliptic boundary value problem (BVP) involving the Paneitz operator with boundary conditions of Chang-Qing and Neumann operators. The corresponding BVP has a variational formulation but the corresponding variational problem, in the…
Consider the following coupled elliptic system of equations \begin{equation*} \label{} (-Δ)^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} \ \ \mathbb{R}^n , \end{equation*} where $0<s\le 2$, $p>1$, $m\ge1$, $u=(u_i)_{i=1}^m$ and $u_i:\mathbb R^n\to \mathbb R$. The qualitative behavior of solutions of the above system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. For the case of local scalar equation, that is when $m=1$ and $s=1$, Gidas and Spruck in \cite{gs} and later Caffarelli, Gidas …
