arXiv research

Study solves a mathematical problem related to elliptic Schroedinger-to-Neumann maps.

Explains mapping properties of elliptic operators in conical spaces.

New framework uses elliptic operators to study projective maps.

Harmonic maps are described using Jacobi elliptic functions.

In the present paper we consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. They are called meridian surfaces of elliptic or hyperbolic type, respectively. We study these surfaces with respec…

Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.

New solutions found for elliptic sinh-Gordon and sine-Gordon equations.

The study proves planes are the only complete uniformly elliptic Weingarten multigraphs.

Develops tools for studying intersections of elliptic operators, focusing on $J$-holomorphic maps.

We study the Abel-Jacobi map for bisections of a certain rational elliptic surface. As an application, we construct examples of Zariski $N$-plets for conic arrangements.

Holomorphic maps between configuration spaces are classified, resolving quartic and elliptic curve problems.

We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…

Characterizes totally elliptic surface group representations into Lie groups.

In this short note we prove that in the case of elliptic curves, the isomorphism of generalized complex structure between $T$-dual manifolds described by Cavalcanti-Gualtieri coincides with the mirror map for elliptic curves described by Polishchuk and Zaslow.

New conservation laws found for polyharmonic maps in critical dimension.

Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.

Method counts zeros of Betti map for elliptic surface sections.

Defines an equivariant index for proper actions by $H imes G$.

We prove that for a fibration of simply-connected spaces of finite type $F\hookrightarrow E\to B$ with $F$ being positively elliptic and $H^*(F,\qq)$ not possessing non-trivial derivations of negative degree, the base $B$ is formal if and only if the total space $E$ is formal. Moreover, in this case the fibration map i…

Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein …

We investigate a parabolic-elliptic system for maps $(u,v)$ from a compact Riemann surface $M$ into a Lorentzian manifold $N\times{\mathbb{R}}$ with a warped product metric. That system turns the harmonic map type equations into a parabolic system, but keeps the $v$-equation as a nonlinear second order constraint along…

New classification for certain 4-manifolds using quasiregular mappings.

Uniform K-homology theory applied to elliptic operators on manifolds with boundary.

The paper studies elliptic surfaces and proves unique fibered structures.

Study confirms nullity of biharmonic maps family, linking spectral and arithmetic geometry.

Functoriality proved for higher rho invariants of elliptic operators.

We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These maps are called affine harmonic. We show an existence result for affine harmonic…

We find a new relation among right-handed Dehn twists in the mapping class group of a $k$-holed torus for $4 \leq k \leq 9$. This relation induces an elliptic Lefschetz pencil structure on the four-manifold \cp $#(9-k)$ \cpb $$ with $k$ base points and twelve singular fibers. By blowing up the base points we get an el…

Constructs maps from field theories to complexified K-theory and elliptic cohomology.

We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scatte…

Study real Mordell-Weil group and real lines on rational elliptic surfaces and del Pezzo surfaces.

For $J$-holomorphic mappings for a strongly pseudo-convex manifold, we prove elliptic regularity by the argument of boots-strapping.

The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.

We study supersymmetric harmonic maps from the point of view of integrable system. It is well known that harmonic maps from R^2 into a symmetric space are solutions of a integrable system . We show here that the superharmonic maps from R^{2|2} into a symmetric space are solutions of a integrable system, more precisely …

On simple geodesic disks of constant curvature, we derive new functional relations for the geodesic X-ray transform, involving a certain class of elliptic differential operators whose ellipticity degenerates normally at the boundary. We then use these relations to derive sharp mapping properties for the X-ray transform…

Using the Plucker map between grassmannians, we study basic aspects of classic grassmannian geometries. For `hyperbolic' grassmannian geometries, we prove some facts (for instance, that the Plucker map is a minimal isometric embedding) that were previously known in the `elliptic' case.

Study moduli spaces of elliptic PDEs using derived $C^{\infty}$-geometry.

Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.

Cube edges curves minimize systole length.

We establish a moduli space $\mathbb E$ of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map $Π$ in $\mathbb E$, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary map $Π$ is Fredholm by showing that the stationary vacuum equations (combined with p…

We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.

We characterize how to vary the Abel-Jacobi map in terms of Schiffer variation. From this characterization, we will interpret the relation of hyperellipticity of curves with Schiffer variation and describe the deformation of elliptic solitons under Schiffer variation.

Study character varieties for 3-punctured sphere group representations in PU(2,1).

Generalizes cohomological obstruction for quasiregular ellipticity.

In this note, we study an invariant associated to the zeros of the moment map generated by an action form, the infinitesimal index. This construction will be used to study the compactly supported equivariant cohomology of the zeros of the moment map and to give formulas for the multiplicity index map of a transversally…

Proposes a new regularization technique for neural networks using elliptic operators.

In this paper, we study rotational surfaces of elliptic, hyperbolic and parabolic type with pointwise 1-type Gauss map which have spacelike profile curve in four dimensional pseudo Euclidean space E4-2 and obtain some characterizations for these rotational surfaces to have pointwise 1-type Gauss map.

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous s…