We give a simpler proof of the sharp Frank-Lieb inequality on the Heisenberg group. The proof bypasses the sophisticated argument for existence of a minimizer and is based on the study of the 2nd variation of subcritical functionals using their fundamental techniques.

Let (M^m,g), m≥2, be a closed connected Riemannian manifold with _g≤-1, and let (X,g_X) be its universal cover. Yau proved that (X)≥ m-1. We prove that equality is rigid: if (X)=m-1, then (X,g_X) is isometric to ^m(-1).

Let (M,g,J) be a closed Kähler manifold satisfying Ric⩾ g. We establish improved Liouville theorems for the Euler–Lagrange equations associated with the Beckner–Sobolev inequalities by incorporating the first positive eigenvalue of the ∂̅-Laplacian into a differential-identity argument. As a consequence, we obtain improved Sobolev and Beckner inequalities that refine the known Riemannian and Kähler estimates when the first eigenvalue is sufficiently large. We also derive new upper bounds for the diameter of (M,g).

Let (X-n,(g +)) be a conformally compact manifold with Ric >=-(n - 1). If g+ is asymptotically Poincar & eacute;-Einstein, we establish a sharp inequality relating the type II Yamabe invariant of X and the Yamabe invariant of its conformal infinity.

In this paper we compute the first and second variation of the normalized Einstein-Hilbert functional on CR manifolds. We characterize critical points as pseudo-Einstein structures. We then turn to the second variation on standard spheres. While the situation is quite similar to the Riemannian case in dimension greater or equal to five, in three dimension we observe a crucial difference, which mainly depends on the embeddable character of the perturbed CR structure.

In this paper we study positive solutions to the CR Yamabe equation in noncompact (2n+1)-dimensional Sasakian manifolds with nonnegative curvature. In particular, we show that the Heisenberg group ℍ^1 is the only (complete) Sasakian space with nonnegative Tanaka-Webster scalar curvature admitting a (nontrivial) positive solution. Moreover, under some natural assumptions, we prove this strong rigidity result in higher dimensions, extending the celebrated Jerison-Lee's result to curved manifolds.

We give a simple proof of a recent result due to Agostiniani, Fogagnolo and Mazzieri.

We revisit some uniqueness results for a geometric nonlinear PDE related to the scalar curvature in Riemannian geometry and CR geometry. In the Riemannian case we give a new proof of the uniqueness result assuming only a positive lower bound for Ricci curvature. We apply the same principle in the CR case and reconstruct the Jerison-Lee identity in a more general setting. As a corollary, we prove a more general uniqueness result for a family of semilinear PDE on a closed pseudohermitian manfiold with zero torsion and positive Ricci curvature. We also discuss some open problems for further study.

For an asymptotically Poincare-Einstein manifold with a lower Ricci curvature bound, we establish a sharp inequality relating the type II Yamabe invariant of the interior and the Yamabe invariant of its conformal infinity

We give a new proof of Aubin's improvement of the Sobolev inequality on $\mathbb{S}^{n}$ under the vanishing of first order moments of the area element and generalize it to higher order moments case. By careful study of an extremal problem on $\mathbb{S}^{n}$, we determine the constant explicitly in the second order moments case.

In this note, we study symmetry of solutions of the elliptic equation -Delta(S2)u+3 = e(2u) on S-2, that arises in the consideration of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the third author. As a consequence, we derive several sharp Sobolev trace inequalities on these manifolds.

We prove some uniqueness results for positive harmonic functions on the unit ball satisfying a nonlinear boundary condition.

We propose to study positive harmonic functions satisfying a nonlinear Neuman condition on a compact Riemannian manifold with nonnegative Ricci curvature and strictly convex boundary. A precise conjecture is formulated. We discuss its implications and present some partial results. Related questions are discussed for compact Riemannian manifolds with positive Ricci curvature and convex boundary.

We generalize Brendle’s geometric inequality considered in Brendle (Publ Math Inst Hautes Études Sci 117:247–269, 2013 ) to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperbolic manifolds in the spirit of Chruściel and Simon (J Math Phys 42(4):1779–1817, 2001 ).

In this paper we prove that two calculus lemmas, which are used in the method of moving sphere for classifying certain constant curvature equation, also hold on Heisenberg group H-n.

We establish an integral formula on a smooth, precompact domain in a Kahler manifold. We apply this formula to study holomorphic extension of CR functions. Using this formula we prove an isoperimetric inequality in terms of a positive lower bound for the Hermitian curvature of the boundary. Combining with a Minkowski type formula on the complex hyperbolic space we prove that any closed, embedded hypersurface of constant mean curvature must be a geodesic sphere, provided the hypersurface is Hopf. A similar result is established on the complex projective space.

On a compact Riemannian manifold with boundary, we study how Ricci curvature of the interior affects the geometry of the boundary. First we establish integral inequalities for functions defined solely on the boundary and apply them to obtain geometric inequalities involving the total mean curvature. Then we discuss related rigidity questions and prove Ricci curvature rigidity results for manifolds with boundary.

We discuss a remarkable formula discovered by Jerison and Lee to classify constant scalar curvature pseudohermitian structures on the sphere. We show that the formula is valid in the wider context of Einstein pseudohermitian manifolds. As an application we prove a uniqueness result that generalizes the theorem of Jerison and Lee.