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In this research, we focus on the symmetry of an ancient solution for a fractional parabolic equation involving logarithmic Laplacian in an entire space. In the process of studying the property of a fractional parabolic equation, we obtained some maximum principles, such as the maximum principle of anti-symmetric function, narrow region principle, and so on. We will demonstrate how to apply these tools to obtain radial symmetry of an ancient solution. [ABSTRACT FROM AUTHOR]

In this paper, we construct an ancient solution by using a given shrinking breather and prove a no shrinking breather theorem for noncompact harmonic Ricci flow under the condition that Sic : = Ric − α ∇ ϕ ⊗ ∇ ϕ is bounded from below. [ABSTRACT FROM AUTHOR]

We survey some ideas regarding the application of the Aleksandrov reflection method in partial differential equation to extrinsic geometric flows of Euclidean hypersurfaces. In this survey, we mention some related and important recent developments of others on the convergence of noncontracting flows and construction and classification of ancient flows.

We derive a matrix version of Li & Yau–type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in [5] for the standard heat equation. We then apply these estimates to obtain some Harnack–type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved.

We derive a matrix version of Li & Yau-type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R. Hamilton did in [5] for the standard heat equation. We then apply these estimates to obtain some Harnack-type inequalities, which give local bounds on the solutions in terms of the geometric quantities involved. [ABSTRACT FROM AUTHOR]

We show a triviality result for "pointwise" monotone in time, bounded "eternal" solutions of the semilinear heat equation $ \begin{equation*} u_{t} = \Delta u + |u|^{p} \end{equation*} $ on complete Riemannian manifolds of dimension $ n \geq 5 $ with nonnegative Ricci tensor, when $ p $ is smaller than the critical Sobolev exponent $ \frac{n+2}{n-2} $.

We use Perron method to prove the existence of ancient solutions of the exterior problem for parabolic k-Hessian equations − u t S k (D 2 u) = 1 with prescribed asymptotic behavior at infinity. [ABSTRACT FROM AUTHOR]

In this short note, polynomial growth ancient solutions to the heat equation of the Hodge Laplacian on a complete manifold are studied. Under the assumption that the manifold has polynomial volume growth, it is shown that such solutions can be explicitly expressed into a polynomial of time t with time independent forms as coefficients. As a result, it is concluded that the dimension of the space of all such solutions with a fixed growth order is necessarily bounded by that of the corresponding space of harmonic forms [ABSTRACT FROM AUTHOR]

We use Perron method to prove the existence of ancient solutions of exterior problem for a kind of parabolic Monge–Ampère equation - u t det D 2 u = f with prescribed asymptotic behavior at infinity outside some certain bowl-shaped domain in the lower half space for n ≥ 3 , where f is a perturbation of 1 at infinity. We raise this problem for the first time and construct a new subsolution to it. We also use similar method to prove the existence of the entire solutions. [ABSTRACT FROM AUTHOR]

We study some qualitative properties of ancient solutions of superlinear heat equations on a Riemannian manifold, with particular interest in positivity and constancy in space. [ABSTRACT FROM AUTHOR]

In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature pinching condition. [ABSTRACT FROM AUTHOR]

We point out that Ecker's local integral quantity agrees with Huisken's global integral quantity at infinity for ancient mean curvature flows if Huisken's one is finite on each time-slice. In particular, this means that the finiteness of Ecker's integral quantity at infinity implies the finiteness of the entropy at infinity. [ABSTRACT FROM AUTHOR]

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation ut = Δu + au log u in complete noncompact Riemannian manifolds. As its application, we show that if u is a positive solution of the equation ut = Δu and log u is of sublinear growth in both spatial and time directions, then u must be constant. This gradient estimate is sharp since it is well known that u(x, t) = ex+t satisfying ut = Δu [ABSTRACT FROM AUTHOR]

As an application of his entropy formula, Perelman (The entropy formula for the Ricci flow and its geometric applications, 2002) proved that every compact shrinking breather solution to the Ricci flow is a shrinking gradient Ricci soliton. Zhang (Asian J Math 18(4):727–756, 2014) and Lu and Zheng (J Geom Anal, 1–7, 2017) proved no shrinking breather theorems in the noncompact case under additional conditions. It is a natural question to ask whether one can generalize Perelman's no shrinking breather theorem to the noncompact case assuming only bounded curvature. This is the result we prove in this paper. Our proof uses Perelman's L -geometry and an idea of Lu and Zheng (J Geom Anal, 1–7, 2017). The novelty of this paper is that we can remove the technical assumptions in Zhang (Asian J Math 18(4):727–756, 2014) and Lu and Zheng (J Geom Anal, 1–7, 2017). [ABSTRACT FROM AUTHOR]

In this paper we generalize the neck-stability theorem of Kleiner–Lott (Acta Math 219: 65–134, 2014) to a special class of four-dimensional nonnegatively curved Type I κ -solutions, namely, those whose asymptotic shrinkers are the standard cylinder S 3 × R . We use this stability result to prove a rigidity theorem: if a four-dimensional Type I κ -solution with nonnegative curvature operator has the standard cylinder S 3 × R as its asymptotic shrinker, then it is exactly the cylinder with its standard shrinking metric. [ABSTRACT FROM AUTHOR]

We investigate the Liouville properties of ancient solutions to the V -harmonic map heat flows from complete noncompact Riemannian manifolds with nonnegative Bakry–Emery Ricci curvature. When the target is a simply connected complete Riemannian manifold with nonpositive sectional curvature, then a Liouville theorem of ancient solutions holds under certain growth condition near infinity. When the target is a complete Riemannian manifold with sectional curvature bounded above by a positive constant, we show that if the image of the ancient solution u is contained in a regular ball near infinity, then u is a constant. [ABSTRACT FROM AUTHOR]

In this article, we consider two different monotone quantities defined for the Ricci flow and show that their asymptotic limits coincide for any ancient solutions. One of the quantities we consider here is Perelman’s reduced volume, while the other is the local quantity discovered by Ecker, Knopf, Ni, and Topping. This establishes a relation between these two monotone quantities. [ABSTRACT FROM AUTHOR]

In [33] , the second author proved Perelman's assertion, namely, for an ancient Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. In this paper, we continue this discussion. It turns out that the curvature operator nonnegativity is not a necessary condition, and we need only to assume a consequence of Hamilton's trace Harnack. Furthermore, we show that this condition holds for steady Ricci solitons with nonnegative Ricci curvature. [ABSTRACT FROM AUTHOR]