Mathematics

In this survey we provide an overview of our recent results concerning Ricci de Turck flow on spaces with isolated conical singularities. The crucial characteristic of the flow is that it preserves the conical singularity. Under certain conditions, Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow. We also discuss the relation to Perelman's entropies in the singular setting, and outline open questions and future reseach directions.

In this short note, we prove a uniqueness result for small entropy self-expanders asymptotic to a fixed cone. This is a direct consequence of the mountain-pass theorem and the integer degree argument proved by J. Bernstein and L. Wang.

In Euclidean 3 -space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigation of surfaces of constant Gaussian curvature K=?? . Such a surface can be constructed from a solution to the Sine-Gordon equation, and vice versa. With this as motivation, by means of the fundamental theorem of surfaces in the Heisenberg group H 1 , we show in this paper that the existence of a constant p -mean curvature surface (without singular points) is equivalent to the existence of a solution to a nonlinear second order ODE, which is a kind of {\bf Liénard equations}. Therefore, we turn to investigate this equation. It is a surprise that we give a complete set of solutions to such ODEs in the p -minimal case, and hence use the types of the solution to divide p -minimal surfaces into several classes. As a result, we obtain a representation of p -minimal surfaces and classify further all p -minimal surfaces. In Section 9, we provide an approach to construct p -minimal surfaces. It turns out that, in some sense, generic p -minimal surfaces can be constructed via this approach. Finally, as a derivation, we recover the Bernstein-type theorem which was first shown in 2005.

We use two non-Riemannian curvature tensors, the ? -curvature and the mean Berwald curvature to characterise a class of Finsler metrics admitting first integrals.

Let M be a smooth manifold. When ? is a group acting on the manifold M by diffeomorphisms one can define the ? -co-invariant cohomology of M to be the cohomology of the differential complex Ω c (M ) ? =span{???γ ???,???Ω c (M),γ?�Γ}. For a Lie algebra G acting on the manifold M , one defines the cohomology of G -divergence forms to be the cohomology of the complex C G (M)=span{ L X ?,???Ω c (M),X?�G}. In this short paper we present a situation where these two cohomologies are infinite dimensional.

Let (M,g) a compact Riemannian n -dimensional manifold with umbilic boundary. It is well know that, under certain hypothesis, in the conformal class of g there are scalar-flat metrics that have the boundary of M as a constant mean curvature hypersurface. In this paper we prove that these metrics are a compact set in the case of low dimensional manifolds, that is n=6,7,8, provided that the Weyl tensor is always not vanishing on the boundary.

Let f be a harmonic map from a Riemann surface to a Riemannian n -manifold. We prove that if there is a holomorphic diffeomorphism h between open subsets of the surface such that f∘h=f , then f factors through a holomorphic map onto another Riemann surface. If such h is anti-holomorphic, we obtain an analogous statement. For minimal maps, this result is well known and is a consequence of the theory of branched immersions of surfaces due to Gulliver-Osserman-Royden. Our proof relies on various geometric properties of the Hopf differential.

We show that two families of equations, the generalized inviscid Proudman-Johnson equation, and the r -Hunter-Saxton equation (recently introduced by Cotter et al.) coincide for a certain range of parameters. This gives a new geometric interpretation of these Proudman-Johnson equations as geodesic equations of right invariant homogeneous W 1,r -Finsler metrics on the diffeomorphism group. Generalizing a construction of Lenells for the Hunter-Saxton equation, we analyze these equations using an isometry from the diffeomorphism group to an appropriate subset of real-valued functions. Thereby we show that the periodic case is equivalent to the geodesic equations on the L r -sphere in the space of functions, and the non-periodic case is equivalent to a geodesic flow on a flat space. This allows us to give explicit solutions to these equations in the non-periodic case, and answer several questions of Cotter et al. regarding their limiting behavior.

In this paper, we provide a sufficient condition for a curve on a surface in R 3 to be given by an orthogonal intersection with a sphere. This result makes it possible to express the boundary condition entirely in terms of the Weierstrass data without integration when dealing with free boundary minimal surfaces in a ball B 3 . Moreover, we show that the Gauss map of an embedded free boundary minimal annulus is one to one. By using this, the Fraser-Li conjecture can be translated into the problem of determining the Gauss map. On the other hand, we show that the Liouville type boundary value problem in an annulus gives some new insight into the structure of immersed minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic approach to the Fraser-Li conjecture.

The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{gradient almost Riemann soliton} in a non-cosymplectic normal almost contact metric manifold M 3 . Before all else, it is proved that if the metric of M 3 is Riemann soliton with divergence-free potential vector field Z , then the manifold is quasi-Sasakian and is of constant sectional curvature - λ , provided α,β= constant. Other than this, it is shown that if the metric of M 3 is \emph{ARS} and Z is pointwise collinear with ξ and has constant divergence, then Z is a constant multiple of ξ and the \emph{ARS} reduces to a Riemann soliton, provided α,β= constant. Additionally, it is established that if M 3 with α,β= constant admits a gradient \emph{ARS} (γ,ξ,λ) , then the manifold is either quasi-Sasakian or is of constant sectional curvature −( α 2 − β 2 ) . At long last, we develop an example of M 3 conceding a Riemann soliton.