arXiv research

Ricci flows with bounded scalar curvature cannot develop Type I singular points.

Proves conditions for positive scalar curvature on certain manifolds with conical singularities.

Constructs Kähler metrics with constant scalar curvature on blown-up manifolds.

Local singularity analysis for Ricci flows with applications to bounded scalar curvature.

New inequalities for austere submanifolds established.

Non-compact flow lines for scalar curvature prescription on manifolds.

It has been showed by Byde that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main…

In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of crit…

The paper extends Ricci flow theory with Type-I scalar curvature bounds, proving entropy convergence and characterizing singular sets.

Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.

In this paper we investigate the properties of small surfaces of Willmore type in Riemannian manifolds. By \emph{small} surfaces we mean topological spheres contained in a geodesic ball of small enough radius. In particular, we show that if there exist such surfaces with positive mean curvature in the geodesic ball $B_…

Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.

Study shows curvature rigidity of specific metric types.

Study geometric structure of Ricci shrinker ends without global curvature assumptions.

We give a general procedure for gluing together possibly noncompact manifolds of constant scalar curvature which satisfy an extra nondegeneracy hypothesis. Our aim is to provide a simple paradigm for making `analytic' connected sums. In particular, we can easily construct complete metrics of constant positive scalar cu…

The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudo gradient flows. W…

The stationary points of the total scalar curvature functional on the space of unit volume metrics on a given closed manifold are known to be precisely the Einstein metrics. One may consider the modified problem of finding stationary points for the volume functional on the space of metrics whose scalar curvature is equ…

We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$-invariant metrics of positive scalar curvature on every $S^1$-manifold which has a fixed point component of codimension 2. As a consequence we can prove that there are non-invariant metr…

Study of exceptional algebroids in type IIA string theory.

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

We show the existence of a local foliation of a three dimensional Riemannian manifold by critical points of the Willmore functional subject to a small area constraint around non-degenerate critical points of the scalar curvature. This adapts a method developed by Rugang Ye to construct foliations by surfaces of constan…

Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.

Study stability of Einstein metrics on homogeneous spaces.

In this paper, we study curvature behavior at the first singular time of solution to the Ricci flow on a smooth, compact n-dimensional Riemannian manifold $M$, $\frac{\partial}{\partial t}g_{ij} = -2R_{ij}$ for $t\in [0,T)$. If the flow has uniformly bounded scalar curvature and develops Type I singularities at $T$, us…

Using holographic renormalization coupled with the Caffarelli/Silvestre\cite{caffarelli} extension theorem, we calculate the precise form of the boundary operator dual to a bulk scalar field rather than just its average value. We show that even in the presence of interactions in the bulk, the boundary operator dual to …

The paper studies metrics on manifolds with scalar curvature properties.

We study positive scalar curvature on the regular part of Riemannian manifolds with singular, uniformly Euclidean ($L^\infty$) metrics that consolidate Gromov's scalar curvature polyhedral comparison theory and edge metrics that appear in the study of Einstein manifolds. We show that, in all dimensions, edge singularit…

Study Nijenhuis operators and their linearization problem using left-symmetric algebras.

New symmetries found for scalar and vector ODEs of arbitrary dimensions.

Mapper and Ball Mapper tools for complex data analysis.

Research characterizes critical points of scalar curvature functionals.

The paper proves properties of Kähler surfaces with zero scalar curvature.

Study of equivariant scalar curvature groups for proper group actions.

Formula for Lichnerowicz Laplacian on invariant metrics, deducing stability of Einstein manifolds.

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy- Riemann spheres under Beta-flatness condition. To give a lower bound for the number of solutions, we use Bahri methods based on the theory of critical points at infinity and a Poincare-Hopf type formula.

Refined estimates for surfaces in curved spaces based on Willmore functional.

We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which…

Let $M$ be a closed manifold of Sasaki type. A polarization of $M$ is defined by a Reeb vector field, and for one such, we consider the set of all Sasakian metrics compatible with it. On this space, we study the functional given by the squared $L^2$-norm of the scalar curvature. We prove that its critical points, or ca…

Study on half spheres solves Nirenberg problem with complex blow-up analysis.

The paper proves conditions under which critical point metrics are Einstein.

Constructs metrics with negative constant scalar curvature.

In this article, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal Kähler metric. In fact, this extremal Kähler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence…

Paper generalizes scalar curvature theorem to weighted manifolds.

The study explores metrics with constant curvature on compact manifolds.

Develops Llarull type theorems for 3D and 4D bands with spectral scalar curvature bounds.

Let $(M,g,φ)$ be a solution to the Ricci flow coupled with the heat equation for a scalar field $φ$. We show that a complete, $κ$-noncollapsed solution $(M,g,φ)$ to this coupled Ricci flow with a Type I singularity at time $T<\infty$ will converge to a non-trivial Ricci soliton after parabolic rescaling, if the base po…

New metrics found on orbifold resolutions with specific curvature.

Study finite time singularities in Ricci flow with bounded scalar curvature.