arXiv research

New Einstein metrics found on a 10-dimensional sphere.

No complex structure exists on a round 6-sphere.

We compute the spectral action of $SU(2)/Γ$ with the trivial spin structure and the round metric and find it in each case to be equal to $\frac{1}{|Γ|} (Λ^3 \hat{f}^{(2)}(0) - 1/4Λ\hat{f}(0))+ O(Λ^{-\infty})$. We do this by explicitly computing the spectrum of the Dirac operator for $SU(2)/Γ$ equipped with the trivial …

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zol…

Eigenvalues on spheres are compared to the unit round sphere, proving a sharp bound and equality condition.

Estimates Bartnik mass for metrics with nonnegative Gauss curvature.

In low dimensions, minimizers for the second conformal eigenvalue do not exist near the round sphere.

Explains visual metrics on hyperbolic space boundaries.

We construct a counterexample to a conjectured inequality L<2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd def…

Theorems prove upper bounds for foliations on closed Alexandrov spaces.

Study shows curvature rigidity of specific metric types.

I review several proofs for non-existence of orthogonal complex structures on the six-sphere, most notably by G. Bor and L. Hernandez-Lamoneda, but also by K. Sekigawa and L. Vanhecke that we generalize for metrics close to the round one. Invited talk at MAM-1 workshop, 27-30 March 2017, Marburg.

New metrics with constant curvature found on spheres minus caps.

Theorem proves spectral rigidity of warped product metrics.

Outer space and Teichmüller space fail well-rounded retract analogy.

In this paper we show that for a generalized Berger metric $\hat{g}$ on $S^3$ close to the round metric, the conformally compact Einstein (CCE) manifold $(M, g)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if $\hat{g}$ is an $\text{SU}(k+1)$-…

We obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These are (complete) constant scalar curvature metrics on the complement of $S^1$ inside $S^m$, $m\geq 5$, that are conformal to the round (incomplete) metric and "periodic" in the sense of…

Hyperkahler manifolds with round Kahler cones have unique bimeromorphic models.

In this paper we show that for a Berger metric $\hat{g}$ on $S^3$, the non-positively curved conformally compact Einstein metric on the $4$-ball $B_1(0)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen \cite{Pedersen}. In particular, since in \ci…

We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with $\mathbb{R}^{2}$. As an application, we show that positively curved metrics on $S^{3}$ and $RP^{3}$ with almost maximal width must be nearly round.

In 1993, Bartnik introduced a quasi-spherical construction of metrics of prescribed scalar curvature on 3-manifolds. Under quasi-spherical ansatz, the problem is converted into the initial value problem for a semi-linear parabolic equation of the lapse function. The original ansatz of Bartnik started with a background …

New contact structures on spheres and tori with weakly compatible metrics.

The Levy-Gromov inequality states that round spheres have the least isoperimetric profile (normalized by total volume) among Riemannian manifolds with a fixed positive lower bound on the Ricci tensor. In this note we study critical metrics corresponding to the Levy-Gromov inequality and prove that, in two-dimensions, t…

Study on positive solutions of Yamabe-type equation on spheres.

We establish several nonuniqueness results for the problem of finding complete conformal metrics with constant (fourth-order) $Q$-curvature on compact and noncompact manifolds of dimension $\geq5$. Infinitely many branches of metrics with constant $Q$-curvature, but without constant scalar curvature, are found to bifur…

The study finds the optimal metrics for free boundary minimal surfaces in spherical caps.

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in $\R^n$ for every $n\geq 9$. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Diric…

Study shows Einstein structures on 4-manifolds are rigid.

Optimizes metrics for the first curl eigenvalue on 3-manifolds.

We prove that the round metric on the sphere has the largest first eigenvalue of the Dirac operator among all metrics that are larger than it. As a corollary, this gives an alternative proof of an extremality result for scalar curvature due to M. Llarull.

We prove that in conformal classes of metrics near the class of an Einstein metric (other than the standard round metric on a sphere) the Yamabe problem has a unique solution up to scaling. This is a local extension, in the space of conformal classes, of a well-known uniqueness criterion due to Obata.

The purpose of this paper is to investigate the critical points of the total scalar curvature functional restricted to space of metrics with constant scalar curvature of unitary volume, for simplicity CPE metrics. It was conjectured in $1980$'s that every CPE metric must be Einstein. We prove that a $4$-dimensional CPE…

Study finds non-uniqueness in sphere metrics with constant fractional curvature.

In this paper we study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimension three and higher and prove that the only complete examples of such metrics on $S^n$, $\R{n}$ and $\R{}\times S^{n-1}$ are, respectively, the round, flat, and standard cylindrical metrics.

Study shows limits of metrics with positive scalar curvature on spheres.

We classify compact conformally flat $n$-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either $\mathbb{S}^{n}$ with the round metric, $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ with the product metric or $\mathbb{S}^{1…

An important tool in the study of conformal geometry, and the AdS/CFT correspondence in physics, is the Fefferman-Graham expansion of conformally compact Einstein metrics. We show that conformally compact metrics satisfying a generalization of the Einstein equation, Poincare-Lovelock metrics, also have Fefferman-Graham…

Curve shortening in metric-affine plane shrinks convex curves to points.

We investigate the low-energy behavior of the gradient flow of the $L^2$ norm of the Riemannian curvature on four-manifolds. Specifically, we show long time existence and exponential convergence to a metric of constant sectional curvature when the initial metric has positive Yamabe constant and small initial energy.

Rigidity theorem for special metrics on 4-manifolds.

The paper examines stability of the Sobolev inequality in metric spaces with curvature dimension conditions.

Upper diameter bound for manifolds with positive scalar curvature.

A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $η$. Several sufficient assumptions on such a surface with non-degenerate $η$-second fundamental form are established to prove that it must be a totally umbilical round sphere. With t…

I prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture stating that on a closed manifold different from the round sphere every projective (i.e., geodesic-preserving) vector field is Killing.

To every real analytic Riemannian manifold M there is associated a complex structure on a neighborhood of the zero section in the real tangent bundle of M. This structure can be uniquely specified in several ways, and is referred to as a Grauert tube. We say that a Grauert tube is entire if the complex structure can be…

We study solutions to the static vacuum Einstein equations on exterior domains with prescribed metric and mean curvature on the inner boundary. It is proved that for any such boundary data near the standard round boundary data in Euclidean space, there exists a unique AF solution to the static vacuum equations realizin…

Study shows how certain curved bundles reduce their structure group.

We prove an important partial case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the round sphere (up to a finite cover).