Non-linear partial differential equations in conformal geometry

In this article we study the nonlocal equation \begin{align} (-\Delta)^{\frac{n}{2}}u=(n-1)!e^{nu}\quad \text{in $\mathbb{R}^n$}, \quad\int_{\mathbb{R}^n}e^{nu}dx<\infty, \notag \end{align} which arises in the conformal geometry. Inspired by the previous work of C. S. Lin and L. Martinazzi in even dimension and T. Jin, A. Maalaoui, L. Martinazzi, J. Xiong in dimension three we classify all solutions to the above equation in terms of their behavior at infinity.

The aim of this paper is to report on recent development on the conformal fractional Laplacian, both from the analytic and geometric points of view, but especially towards the PDE community.

The conformal powers of the Laplacian of a Riemannian metric which are known as the GJMS-operators admit a combinatorial description in terms of the Taylor coefficients of a natural second-order one-parameter family $\H(r;g)$ of self-adjoint elliptic differential operators. $\H(r;g)$ is a non-Laplace-type perturbation of the conformal Laplacian $P_2(g) = \H(0;g)$. It is defined in terms of the metric $g$ and covariant derivatives of the curvature of $g$. We study the heat ker...

In this paper we prescribe a fourth order conformal invariant on the standard $n-$sphere, with $n\geq5$, and study the related fourth order elliptic equation. We first find some existence results in the perturbative case. After some blow up analysis we build a homotopy to pass from the perturbative case to the non-perturbative one under some flatness condition. Finally we state some existence results under the assumption of symmetry.

We solve the $\sigma_2$-Yamabe problem for a non locally conformally flat manifold of dimension $n>8$.

We introduce a double iterative scheme and local variational method to solve the Yamabe-type equation $ - \frac{4(n - 1)}{n - 2}\Delta_{g} u + (S_{g} + \beta ) u = \lambda u^{\frac{n + 2}{n - 2}} $ for some constant $ \beta \leqslant 0 $, locally on Riemannian domain $ (\Omega, g) $ with trivial Dirichlet condition and globally on closed manifolds $ (M, g) $; the dimensions of $ \Omega $ and $ M $ are at least 3. In contrast to the traditional global variational method, these...

We propose and discuss recursive formulas for conformally covariant powers $P_{2N}$ of the Laplacian (GJMS-operators). For locally conformally flat metrics, these describe the non-constant part of any GJMS-operator as the sum of a certain linear combination of compositions of lower order GJMS-operators (primary part) and a second-order operator which is defined by the Schouten tensor (secondary part). We complete the description of GJMS-operators by proposing and discussing r...

We investigate second order quasilinear equations of the form f_{ij} u_{x_ix_j}=0 where u is a function of n independent variables x_1, ..., x_n, and the coefficients f_{ij} are functions of the first order derivatives p^1=u_{x_1}, >..., p^n=u_{x_n} only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n+1, R), which acts by projective transformations on the space P^n with coordinates p^1, ..., p^n. The coefficient matrix f_{ij} defines on...

We define a new formal Riemannian metric on a conformal class in the context of the $v_{\frac{n}{2}}$-Yamabe problem. Our construction leads to a new variational characterization and a new parabolic flow approach to this problem. Moreover, this variational framework suggests that solutions to this problem are unique in a given conformal class.

In the lecture notes, the author will survey the development of conformal geometry on four dimensional manifolds. The topic she chooses is one on which she has been involved in the past twenty or more years: the study of the integral conformal invariants on 4-manifolds and geometric applications. The development was heavily influenced by many earlier pioneer works; recent progress in conformal geometry has also been made in many different directions, here we will only present...