A5025588918- Geometric Analysis and Curvature Flows
- Geometry and complex manifolds
- Geometric and Algebraic Topology
- Advanced Differential Geometry Research
- Point processes and geometric inequalities
- Nonlinear Partial Differential Equations
- Black Holes and Theoretical Physics
- Advanced Operator Algebra Research
- Algebraic and Geometric Analysis
- Mathematics and Applications
- Mathematical Dynamics and Fractals
- Advanced Numerical Analysis Techniques
- Advanced Mathematical Physics Problems
- Cosmology and Gravitation Theories
- Computational Geometry and Mesh Generation
- Inflammatory Myopathies and Dermatomyositis
- Analytic and geometric function theory
- Advanced Differential Equations and Dynamical Systems
- Numerical methods in inverse problems
- Advanced Mathematical Theories and Applications
- Advanced Materials and Mechanics
- Advanced Numerical Methods in Computational Mathematics
- Digital Image Processing Techniques
- Relativity and Gravitational Theory
Michigan State University
2025
Duke University
2022-2024
Stony Brook University
2016-2022
Simons Center for Geometry and Physics
2018-2019
University of Oregon
2014-2015
We give a lower bound for the Lorentz length of ADM energy-momentum vector (ADM mass) 3‑dimensional asymptotically flat initial data sets Einstein equations. The is given in terms linear growth 'spacetime harmonic functions' addition to density matter fields, and valid regardless whether dominant energy condition holds or possess boundary. A corollary this result new proof spacetime positive mass theorem complete those with weakly trapped surface boundary, includes rigidity statement which...
We produce new examples of Riemannian manifolds with scalar curvature lower bounds and collapsing behavior along codimension 2 submanifolds. Applications this construction are given, primarily on questions concerning the stability rigidity phenomena, such as Llarull's Theorem Positive Mass Theorem.
Abstract Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms positive pointwise lower scalar curvature, assuming certain topological conditions. We extend several incarnations these results which curvature are replaced spectral bounds. More precisely, we prove principal eigenvalue operator $-\Delta +cR$, where $R$ denotes and $c>0$ is a constant. Three separate strategies employed to obtain distinct holding...
We prove a differential Harnack inequality for the Endangered Species Equation, which is nonlinear parabolic equation. Our derivation relies on an idea related to maximum principle. As application of this inequality, we will show that positive solutions equation must blow up in finite time.
We give an explicit formula for all curves of constant torsion in the unit two-sphere.Our approach uses hypergeometric functions to solve relevant ordinary differential equations.
In the pioneering work of Stern, level sets harmonic functions have been shown to be an effective tool in study scalar curvature dimension 3. Generalizations this idea, utilizing so called spacetime as well other elliptic equations, are similarly treating geometric inequalities involving ADM mass. paper, we survey recent results context, focusing on applications asymptotically flat and hyperbolic versions positive mass theorem, additionally introduce a new concept total valid both settings...
We show that the periodic η-invariant of Mrowka, Ruberman and Saveliev provides an obstruction to existence cobordisms with positive scalar curvature metrics between manifolds dimensions 4 6.Our proof combines end-periodic index theorem a relative version Schoen-Yau minimal surface technique.As result, we bordism groups Ω spin,+ n+1 (S 1 × BG) are infinite for any non-trivial group G which is fundamental spin spherical space form dimension n = 3 or 5. 870 D. Kazaras, Ruberman, N. infinitely...
Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and contains variety which provide sharp relationships between this notions {\em{width}}. Some inequalities leverage quantities such as boundary mean while others involve topological conditions in form linking requirements homological In several these results open...
Work of D. Stern and Bray-Kazaras-Khuri-Stern provide differential-geometric identities which relate the scalar curvature Riemannian 3-manifolds to global invariants in terms harmonic functions. These quantitative formulas are useful for stability results show promise more applications this type. In paper, we analyze maps flat model spaces order address conjectures concerning geometric positive mass theorem Geroch conjecture. By imposing integral Ricci isoperimetric bounds, leverage...
We give a lower bound for the Lorentz length of ADM energy-momentum vector (ADM mass) 3-dimensional asymptotically flat initial data sets Einstein equations. The is given in terms linear growth `spacetime harmonic functions' addition to density matter fields, and valid regardless whether dominant energy condition holds or possess boundary. A corollary this result new proof spacetime positive mass theorem complete those with weakly trapped surface boundary, includes rigidity statement which...
We show that the periodic $\eta$-invariants introduced by Mrowka--Ruberman--Saveliev~\cite{MRS3} provide obstructions to existence of cobordisms with positive scalar curvature metrics between manifolds dimensions $4$ and $6$. The proof combines a relative version Schoen--Yau minimal surface technique an end-periodic index theorem for Dirac operator. As result, we bordism groups $\Omega^{spin,+}_{n+1}(S^1 \times BG)$ are infinite any non-trivial group $G$ which is fundamental spin spherical...
We introduce the notion of a smocked metric space and explore balls geodesics in collection different spaces. find their rescaled Gromov-Hausdorff limits prove these tangent cones at infinity exist, are unique, normed close with variety open questions suitable for advanced undergraduates, masters students, doctoral students.
We establish Gromov-Hausdorff stability of the Riemannian positive mass theorem under assumption a Ricci curvature lower bound. More precisely, consider class orientable complete uniformly asymptotically flat 3-manifolds with nonnegative scalar curvature, vanishing second homology, and uniform bound on curvature. prove that if sequence such manifolds has ADM approaching zero, then it must converge to Euclidean 3-space in pointed sense. In particular, this confirms Huisken Ilmanen's...
We prove a differential Harnack inequality for the Endangered Species Equation, nonlinear parabolic equation. Our derivation relies on an idea related to maximum principle. As application of this inequality, we will show that positive solutions equation must blowup in finite time. also use partially answer question Hamilton, 2011.
We establish a gluing theorem for solutions of Yamabe problem manifolds with boundary studied by Escobar in the 90's. Given two scalar-flat Riemannian whose has zero mean curvature and sharing submanifold $K$, we produce generalized connected sum along $K$. On this third manifold family metrics small, constant on which are close to original $C^2$ sense. Under extra geometric conditions manifolds, can arrange also have vanishing boundary.