Carlo Mantegazza

Motion by curvature of planar networks

We prove that the curvature flow of an embedded planar network of three curves connected through a triple junction, with fixed endpoints on the boundary of a given strictly convex domain, exists smooth until the lengths of the three curves stay far from zero. If this is the case for all times, then the evolution exists for all times and the network converges to the Steiner minimal connection between the three endpoints.

Curvature and distance function from a manifold

This paper is concerned with the relations between the differential invariants of a smooth manifold embedded in the Euclidean space and the square of the distance function from the manifold. In particular, we are interested in curvature invariants like the mean curvature vector and the second fundamental form. We find that these invariants can be computed in a very simple way using the third order derivatives of the squared distance function. Moreover, we study a general class of functionals depending on the derivatives up to a given order γ of the squared distance function and we find an algorithm for the computation of the Euler equation. Our class of functionals includes as particular cases the well-known area functional (γ = 2), the integral of the square of the quadratic norm of the second fundamental form (γ = 3), and the Willmore functional.

Lecture notes on mean curvature flow

Foreword.- Chapter 1. Definition and Short Time Existence.- Chapter 2. Evolution of Geometric Quantities.- Chapter 3. Monotonicity Formula and Type I Singularities.- Chapter 4. Type II Singularities.- Chapter 5. Conclusions and Research Directions.- Appendix A. Quasilinear Parabolic Equations on Manifolds.- Appendix B. Interior Estimates of Ecker and Huisken.- Appendix C. Hamiltons Maximum Principle for Tensors.- Appendix D. Hamiltons Matrix Li-Yau-Harnack Inequality in Rn.- Appendix E. Abresch and Langer Classification of Homothetically Shrinking Closed Curves.- Appendix F. Important Results without Proof in the Book.- Bibliography.- Index.

The Evolution of the Weyl Tensor under the Ricci Flow@@@L’évolution du tenseur de Weyl d’une variété par le flot de Ricci

We compute the evolution equation of the Weyl tensor under the Ricci flow of a Riemannian manifold and we discuss some consequences for the classification of locally conformally flat Ricci solitons.

Locally conformally flat quasi-Einstein manifolds

Abstract In this paper we prove that any complete locally conformally flat quasi-Einstein manifold of dimension n ≧ 3 is locally a warped product with (n − 1)-dimensional fibers of constant curvature. This result includes also the case of locally conformally flat gradient Ricci solitons.

ON THE GLOBAL STRUCTURE OF CONFORMAL GRADIENT SOLITONS WITH NONNEGATIVE RICCI TENSOR

In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product ℝ × Nn-1, or globally conformally equivalent to the Euclidean space ℝn or to the round sphere 𝕊n. In particular, we show that any complete, noncompact, gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric, whenever the potential function is nonconstant.

On some notions of tangent space to a measure

We consider some definitions of tangent space to a Radon measure μ on IR which have been given in the literature. In particular we focus our attention on a recent distributional notion of tangent vector field to a measure and we compare it to other definitions coming from Geometric Measure Theory, based on the idea of blow–up. After showing some classes of examples, we prove an estimate from above for the dimension of the tangent spaces and a rectifiability theorem which also includes the case of measures supported on sets of variable dimension.

Flow by mean curvature inside a moving ambient space

We show some computations related to the motion by mean curvature flow of a submanifold inside an ambient Riemannian manifold evolving by Ricci or backward Ricci flow. Special emphasis is given to the possible generalization of Huisken’s monotonicity formula and its connection with the validity of some Li–Yau–Hamilton differential Harnack-type inequalities in a moving Riemannian manifold.

Locally conformally flat ancient Ricci flows

We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature. In this paper, we study ancient solutions to the Ricci flow. We recall that a time-dependent metric g(t) on a Riemannian manifold M is a solution to the Ricci flow if it evolves by the equation @ @t g(t) = −2Ric g(t) . A solution is called ancient if it is defined for every negative time. Ancient solutions typi- cally arise as the limit of a sequence of suitable blow-ups as the time approaches a singular time for the Ricci flow. In dimension two there exists a compact, rotationally symmetric, ancient solution due to King (22), Rosenau (32) and Fateev, Onofri and Zamolodchikov (18). In dimension three Perelman (30) constructed a compact, rotationally symmetric, ancient solution on the three sphere. In the non-rotationally symmetric case, the first construction is due to Fateev (17) in dimension three. Motivated by this construction, Bakas, Kong and Ni (3) produced high dimensional compact ancient solutions to the Ricci flow which are not rotationally symmetric. In dimension two, Daskalopoulos, Hamilton and Sesum (15) have obtained a complete classification of all compact ancient solutions to the Ricci flow. Ni (27) showed that any com- pact ancient solution to the Ricci flow which is of type I, is k-noncollapsed, and has positive curvature operator has constant sectional curvature. In (6) Brendle, Huisken and Sinestrari proved that any compact ancient solution which satisfies a suitable pinching condition must have constant sectional curvature. In this article, we show that any complete ancient solution to the Ricci flow in dimension n ≥ 4 which is locally conformally flat along the flow must be rotationally symmetric. Theorem 1.1. Let (M n ,g(t)), n ≥ 4, be a complete ancient solution to the Ricci flow which is locally conformally flat at every time. Then (M n ,g(t)) is rotationally symmetric. The non-rotationally symmetric examples of Bakas, Kong and Ni show that the locally conformally flatness assumption cannot be removed. The proof of Theorem 1.1 relies on a previous work of the first two authors (11) about the behavior of the Weyl tensor under the

On the distributional Hessian of the distance function

We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. At the same time we discuss some geometrical properties of the cut locus of a point, and compare some different weak notions of the Hessian and Laplacian.