arXiv research

Proves Schoen's conjecture on tori with specific conditions.

A minimal hypersurface in a sphere is uniquely determined.

The paper confirms a conjecture for 3D manifolds and extends it to 3-7D under specific conditions.

The paper proves foliation of area-minimizing hypersurfaces in asymptotically flat manifolds.

In this short paper, we will give a simple and transcendental proof for Mok's theorem of the generalized Frankel conjecture. This work is based on the maximum principle in \cite{BS2} proposed by Brendle and Schoen.

Proves properties of 4-manifolds with scalar curvature constraints.

We prove that given a hyperbolic manifold endowed with an auxiliary Riemannian metric whose sectional curvature is negative and whose volume is sufficiently small in comparison to the hyperbolic one, we can always find for any radius at least $1$ a ball in its universal cover whose volume is bigger than the hyperbolic …

We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones $C_{p,q}$ are obstructions to the existence problems of special Lagrangian…

We prove that a quasiconformal map of the 2-sphere admits a harmonic quasi-isometric extension to the 3-dimensional hyperbolic space, thus confirming the well known Schoen Conjecture in dimension 3.

We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.

A classical result of Sampson and Schoen-Yau in 1978 states that every diffeomorphism between compact hyperbolic Riemann surfaces is homotopic to an harmonic diffeomorphism. As conjectured by Schoen in 1993 and partially proved by Wan in 1992 and Tam-Wan in 1995, we prove in this article that this theorem generalizes t…

We prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two. This confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.

We show that a Riemannian $3$-manifold with non-negative scalar curvature is flat if it contains an area-minimizing cylinder. This scalar-curvature analogue of the classical splitting theorem of J.~Cheeger and D.~Gromollhas been conjectured by D.~Fischer-Colbrie and R.~Schoen and by M.~Cai and G.~Galloway.

Doing surgery on the 5-torus, we construct a 5-dimensional closed spin-manifold M with $π_1(M) = Z^4times Z/3$, so that the index invariant in the KO-theory of the reduced $C^*$-algebra of $π_1(M)$ is zero. Then we use the theory of minimal surfaces of Schoen/Yau to show that this manifolds cannot carry a metric of pos…

New non-perturbative counterexamples to Min-Oo's Conjecture are created.

The following version of a conjecture of Fischer-Colbrie and Schoen is proved: If M is a complete Riemannian 3-manifold with nonnegative scalar curvature which contains a two-sided torus S which is of least area in its isotopy class then M is flat. This follows from a local version derived in the paper.

We prove the following comparison theorem for metrics with nonnegative scalar curvature, also known as the dihedral rigidity conjecture by Gromov: for $n\le 7$, if an $n$-dimensional prism has nonnegative scalar curvature and weakly mean convex faces, then its dihedral angle cannot be everywhere not larger than its Euc…

In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool …

Proves non-existence of metrics with positive curvature for certain connected sums.

For each integer $g\geq 1$ we use variational methods to construct in the unit $3$-ball $B$ a free boundary minimal surface $Σ_g$ of symmetry group $\mathbb{D}_{g+1}$. For $g$ large, $Σ_g$ has three boundary components and genus $g$. As $g\rightarrow\infty$ the surfaces $Σ_g$ converge as varifolds to the union of the d…

Proves mass theorem for AF manifolds with conical singularities.

In this article, we investigate the volume comparison with respect to scalar curvature. In particular, we show volume comparison holds for small geodesic balls of metrics near a V-static metric. For closed manifold, we prove the volume comparison for metrics near a strictly stable Einstein metric. As applications, we g…

Discuss Alan Schoen's I-WP minimal surface with geometric realizations.

In this paper, I shall demonstrate that sufficiently high-dimensional closed positively-curved Riemannian manifolds are either diffeomorphic to a spherical space form, or isometric to a locally compact rank one symmetric space. This surprising classification of positively-curved Riemannian manifolds results from combin…

Proves mass theorem for manifolds with arbitrary ends.

In this paper we develop new methods for studying the convergence problem for the heat flow on negatively curved spaces and prove that any quasiconformal map of the sphere $\mathbb{S}^{n-1}$, $n\geq 3$, can be extended to the $n$-dimensional hyperbolic space such that the heat flow starting with this extension converge…

We review the Carlotto-Schoen construction of general relativistic initial data sets which are trivial outside of cones, discuss the context, the implications, and some further developments.

We carry out a Carlotto-Schoen-type gluing with interpolating scalar curvature on cone-like sets, or deformations thereof, in the category of smooth Riemannian asymptotically Euclidean metrics.

We study complete minimal graphs in HxR, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in H. We give necessary and sufficient conditions on the "lenghts" of the sides of the polygon (and all inscribed polygons in Γ) that ensure the existence…

The Gromov-Lawson-Rosenberg-conjecture for a group G states that a closed spin manifold M^n (n>4) with fundamental group G admits a metric with positive scalar curvature if and only if its C^*-index A(M) in KO_n(C^*_r(G)) vanishes. We prove this for groups G with low-dimensional classifying space, provided the assembly…

The Positive Mass Conjecture states that any complete asymptotically flat manifold of nonnnegative scalar curvature has nonnegative mass. Moreover, the equality case of the Positive Mass Conjecture states that in the above situation, if the mass is zero, then the Riemannian manifold must be Euclidean space. The Positiv…

Constructs surfaces with conical singularities using variational methods.

Let $Σ$ be a compact Riemann surface and $D_1,...,D_n$ a finite number of pairwise disjoint closed disks of $Σ$. We prove the existence of a proper harmonic map into the Euclidean plane from a hyperbolic domain $Ω$ containing $Σ\backslash\cup_{j=1}^n D_j$ and of its topological type. Here, $Ω$ can be chosen as close as…

Paper presents a new Pohozaev-Schoen identity for non-compact manifolds.

The study finds a continuous map achieving minmax area under Legendrian constraints.

The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.

Schoen-Webster theorem asserts a pseudoconvex CR manifold whose automorphism group acts non properly is either the standard sphere or the Heisenberg space. The purpose of this paper is to survey successive works around this result and then provide a short geometric proof in the compact case.

In this note we show how a generalized Pohozaev-Schoen identity due to Gover and Orsted \cite{GO} can be used to obtain some rigidity results for $V$-static manifolds and generalized solitons. We also obtain an Alexandrov type result for certain hypersurfaces in Einstein manifolds.

Volume comparison theorem for rank 1 symmetric spaces proved.

We prove the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or the total space of an orbifiber bundle over $\mathbb{S}^1$ or $\mathcal{I}$ with generic fiber diffeomorphic to $\mathbb{S}^{n-1}/Γ$ such …

By extending and generalising previous work by Ros and Savo, we describe a method to show that the Morse index of every closed minimal hypersurface on certain positively curved ambient manifolds is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a r…

The study of stable minimal surfaces in Riemannian $3$-manifolds $(M, g)$ with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when $(M, g)$ is asymptotically flat and has horizon boundary. As a consequence, we obtain an effective version of the positive mass theorem …

Schoen-Yau's zero mass theorem stability remains an open question.

Maps with many singularities found in complex space.

In this paper, we study the shape of the min-max minimal hypersurface produced by Almgren-Pitts-Schoen-Simon \cite{AF62, AF65, P81, SS81} in a Riemannian manifold $(M^{n+1}, g)$ of positive Ricci curvature for all dimensions. The min-max hypersurface has a singular set of Hausdorff codimension $7$. We characterize the …

Continuous metrics on manifolds with singularities are shown to be Einstein.

We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension $n=3$ to dimensions $3 \leq n <8$. This requires us to address several technical difficulties that are not present when $n=3$. The regularity and decay assumptions for the initial data sets to which our argume…

The classical Pohozaev identity constrains potential solutions of certain semilinear PDE boundary value problems. The Kazdan-Warner identity is a similar necessary condition important for the Nirenberg problem of conformally prescribing scalar curvature on the sphere. For dimensions $n\geq 3$ both identities are captur…