## Boundary-value problems on $\mathbb{S}^n$ for surfaces of constant Gauss curvature

Pages 601-624 by Bo Guan, Joel Spruck | From volume 138-3

## Regularity estimates for the oblique derivative problem

Pages 1-70 by Pengfei Guan, Eric Sawyer | From volume 137-1

## Convex hypersurfaces of prescribed curvature

Pages 655-673 by Bo Guan, Pengfei Guan | From volume 156-2

## The extremal function associated to intrinsic norms

Through the study of the degenerate complex Monge-Ampère equation, we establish the optimal regularity of the extremal function associated to intrinsic norms of Chern-Levine-Nirenberg and Bedford-Taylor. We prove a conjecture of Chern-Levine-Nirenberg on the extended intrinsic norms on complex manifolds and verify Bedford-Taylor’s representation formula for these norms in general.

Pages 197-211 by Pengfei Guan | From volume 156-1

## A proof of Demailly’s strong openness conjecture

In this article, we solve the strong openness conjecture on the multiplier ideal sheaf associated to any plurisubharmonic function, which was posed by Demailly.

Pages 605-616 by Qi'an Guan, Xiangyu Zhou | From volume 182-2

## A solution of an $L^{2}$ extension problem with an optimal estimate and applications

In this paper, we prove an $L^2$ extension theorem with an optimal estimate in a precise way, which implies optimal estimate versions of various well-known $L^2$ extension theorems. As applications, we give proofs of a conjecture of Suita on the equality condition in Suita’s conjecture, the so-called $L$-conjecture, and the extended Suita conjecture. As other applications, we give affirmative answer to a question by Ohsawa about limiting case for the extension operators between the weighted Bergman spaces, and we present a relation of our result to Berndtsson’s important result on log-plurisubharmonicity of the Bergman kernel.

Pages 1139-1208 by Qi'an Guan, Xiangyu Zhou | From volume 181-3

## Construction of Cauchy data of vacuum Einstein field equations evolving to black holes

We show the existence of complete, asymptotically flat Cauchy initial data for the vacuum Einstein field equations, free of trapped surfaces, whose future development must admit a trapped surface. Moreover, the datum is exactly a constant time slice in Minkowski space-time inside and exactly a constant time slice in Kerr space-time outside.
The proof makes use of the full strength of Christodoulou’s work on the dynamical formation of black holes and Corvino-Schoen’s work on the construction of initial data sets.

Pages 699-768 by Junbin Li, Pin Yu | From volume 181-2

## Multiplicity one theorems: the Archimedean case

Let $G$ be one of the classical Lie groups $\mathrm{GL}_{n+1}(\mathbb{R})$, $\mathrm{GL}_{n+1}(\mathbb{C})$, $\mathrm{U}(p,q+1)$, $\mathrm{O}(p,q+1)$, $\mathrm{O}_{n+1}(\mathbb{C})$, $\mathrm{SO}(p,q+1)$, $\mathrm{SO}_{n+1}(\mathbb{C})$, and let $G’$ be respectively the subgroup $\mathrm{GL}_{n}(\mathbb{R})$, $\mathrm{GL}_{n}(\mathbb{C})$, $\mathrm{U}(p,q)$, $\mathrm{O}(p,q)$, $\mathrm{O}_n(\mathbb{C})$, $\mathrm{SO}(p,q)$, $\mathrm{SO}_n(\mathbb{C})$, embedded in $G$ in the standard way. We show that every irreducible Casselman-Wallach representation of $G’$ occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of $G$. Similar results are proved for the Jacobi groups $\mathrm{GL}_{n}(\mathbb{R})\ltimes \mathrm{H}_{2n+1}(\mathbb{R})$, $\mathrm{GL}_{n}(\mathbb{C})\ltimes \mathrm{H}_{2n+1}(\mathbb{C})$, $\mathrm{U}(p,q)\ltimes \mathrm{H}_{2p+2q+1}(\mathbb{R})$, $\mathrm{Sp}_{2n}(\mathbb{R})\ltimes \mathrm{H}_{2n+1}(\mathbb{R})$, $\mathrm{Sp}_{2n}(\mathbb{C})\ltimes \mathrm{H}_{2n+1}(\mathbb{C})$, with their respective subgroups $\mathrm{GL}_{n}(\mathbb{R})$, $\mathrm{GL}_{n}(\mathbb{C})$, $\mathrm{U}(p,q)$, $\mathrm{Sp}_{2n}(\mathbb{R})$, and $\mathrm{Sp}_{2n}(\mathbb{C})$.

Pages 23-44 by Binyong Sun, Chen-Bo Zhu | From volume 175-1

## Canonical subgroups of Barsotti-Tate groups

Let $S$ be the spectrum of a complete discrete valuation ring with fraction field of characteristic $0$ and perfect residue field of characteristic $p\geq 3$. Let $G$ be a truncated Barsotti-Tate group of level $1$ over $S$. If “$G$ is not too supersingular”, a condition that will be explicitly expressed in terms of the valuation of a certain determinant, then we prove that we can canonically lift the kernel of the Frobenius endomorphism of its special fiber to a subgroup scheme of $G$, finite and flat over $S$. We call it the canonical subgroup of $G$.

Pages 955-988 by Yichao Tian | From volume 172-2

## Geodesic flows with positive topological entropy, twist maps and hyperbolicity

We prove a perturbation lemma for the derivative of geodesic flows in high dimension. This implies that a $C^2$ generic riemannian metric has a nontrivial hyperbolic basic set in its geodesic flow.

Pages 761-808 by Gonzalo Contreras | From volume 172-2

## A generic property of families of Lagrangian systems

We prove that a generic Lagrangian has finitely many minimizing measures for every cohomology class.

Pages 1099-1108 by Patrick Bernard, Gonzalo Contreras | From volume 167-3

## Isometric actions of simple Lie groups on pseudoRiemannian manifolds

Let $M$ be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group $G$. If $m_0, n_0$ are the dimensions of the maximal lightlike subspaces tangent to $M$ and $G$, respectively, where $G$ carries any bi-invariant metric, then we have $n_0 \leq m_0$. We study $G$-actions that satisfy the condition $n_0 = m_0$. With no rank restrictions on $G$, we prove that $M$ has a finite covering $\widehat{M}$ to which the $G$-action lifts so that $\widehat{M}$ is $G$-equivariantly diffeomorphic to an action on a double coset $K\backslash L/\Gamma$, as considered in Zimmer’s program, with $G$ normal in $L$ (Theorem A). If $G$ has finite center and $\mathrm{rank}_{\mathbb{R}}(G)\geq 2$, then we prove that we can choose $\widehat{M}$ for which $L$ is semisimple and $\Gamma$ is an irreducible lattice (Theorem B). We also prove that our condition $n_0 = m_0$ completely characterizes, up to a finite covering, such double coset $G$-actions (Theorem C). This describes a large family of double coset $G$-actions and provides a partial positive answer to the conjecture proposed in Zimmer’s program.

Pages 941-969 by Raul Quiroga-Barranco | From volume 164-3

## Corrigendum to “Convex hypersurfaces of prescribed curvature”

Pages 1099-1099 by Bo Guan, Pengfei Guan | From volume 158-3