arXiv research

Classifies charge-3 monopoles with symmetry, identifying new spectral curves.

Study shows conditions for rational ellipticity of manifolds with symmetries.

The paper introduces elliptic quasi-modular forms via moduli spaces.

We find a remarkable subalgebra of higher symmetries of the elliptic Euler-Darboux equation. To this aim we map such equation into its hyperbolic analogue already studied by Shemarulin. Taking into consideration how symmetries and recursion operators transform by this complex contact transformation, we explicitly give …

Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.

Researchers prove mirror symmetry for certain non-compact Calabi-Yau surfaces.

Given the Lagrangian fibration $T^4\to T^2$ and a Lagrangian submanifold, exhibiting an elliptic umbilic and supporting a flat line bundle, we study, in the context of mirror symmetry, the ``quantum'' corrections necessary to solve the monodromy of the holomorphic structure of the mirror bundle on the dual fibration.

We prove several results about the vanishing of the elliptic genus on positively curved Spin manifolds with logarithmic symmetry rank. The proofs are based on the rigidity of the elliptic genus and Kennard's improvement of the Connectedness Lemma for transversely intersecting, totally geodesic submanifolds.

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.

We study fixed points of smooth torus actions on closed manifolds using fixed point formulas and equivariant elliptic genera. We also give applications to positively curved Riemannian manifolds with symmetry.

We introduce self-dual manifolds and show that they can be used to encode mirror symmetry for affine-Kähler manifolds and for elliptic curves. Their geometric properties, especially the link with special lagrangian fibrations and the existence of a transformation similar to the Fourier-Mukai functor, suggest that this …

This is the first part of a series of papers where we compute Euler characteristics, signatures, elliptic genera, and a number of other invariants of smooth manifolds that admit Riemannian metrics with positive sectional curvature and large torus symmetry. In the first part, the focus is on even-dimensional manifolds i…

Let $x\in\mathbb{R}^{n}$. For $φ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $φ^{t}=t^{-1}φ(xt)$. A projective flow is a solution to the projective translation equation $φ^{t+s}=φ^{t}\circφ^{s}$, $t,s\in\mathbb{R}$. Previously we have developed an arithmetic, topologic and analytic theory of $2$-d…

Elliptic systems are characterized by Darboux integrability.

The main aim of this paper is to study soliton surfaces immersed in Lie algebras associated with ordinary differential equations (ODE's) for elliptic functions. That is, given a linear spectral problem for such an ODE in matrix Lax representation, we search for the most general solution of the wave function which satis…

We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We al…

Elliptic curve governs Hopf linking in symmetric tensegrity.

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…

New guarantees for VI in symmetric cases, extending previous results.

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instan…

Study describes global existence and convergence of flows on surfaces and fibrations.

We introduce a new elliptic operator on null hypersurfaces of four-dimensional Lorentzian manifolds. This operator depends on the first and second fundamental forms of the sections of a foliation of the null hypersurface and its novelty originates from its covariant transformation under change of foliation. It thus pro…

Simply-connected manifolds of positive sectional curvature $M$ are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipt…

Researchers find shortest paths on a special group structure.

The elliptic Hall algebra governs torus link homology.

Proves SYZ mirror symmetry for del Pezzo and rational elliptic surfaces.

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.

A compact complex manifold is Kobayashi non-hyperbolic if there exists an entire curve on it. Using mirror symmetry we establish that there are (possibly singular) elliptic or rational curves on any Calabi-Yau manifold $X$, whose mirror dual $\check X$ exists and is not "Hodge degenerate", therefore proving that $X$ is…

Study elliptic Weingarten surfaces in warped product space with specific curvature conditions.

We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, w…

We consider least energy solutions to the nonlinear equation $-Δ_g u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$ which include the classical hyperbolic space $\mathbb H^n$ as well as manifolds with unbounded sectional geometry. Partial symmetry and existence of least energy solutions is …

We study elliptic gradient systems with fractional laplacian operators on the whole space $(- Δ)^\mathbf s \mathbf u =\nabla H (\mathbf u) \ \ \text{in}\ \ \mathbf{R}^n,$ where $\mathbf u:\mathbf{R}^n\to \mathbf{R}^m$, $H\in C^{2,γ}(\mathbf{R}^m)$ for $γ> \max(0,1-2\min \left \{s_i \right \})$, $\mathbf s=(s_1,\cdot…

In this paper we study the geometry and the topology of unbounded domains in the Hyperbolic Space $\mathbb{H} ^n$ supporting a bounded positive solution to an overdetermined elliptic problem. Under suitable conditions on the elliptic problem and the behaviour of the bounded solution at infinity, we are able to show tha…

We prove that a Lefschetz fibration over the disc that, after compactification, has the same singular fibers as an extremal rational elliptic surface can be obtained by deleting a singular fiber and a section from the rational extremal elliptic surface, i.e. such a Lefschetz fibration is determined up to topological eq…

In this paper we show that a simply connected 8-dimensional manifold M of positive sectional curvature and symmetry rank $\geq 2$ resembles a rank one symmetric space in several ways. For example, the Euler characteristic of M is equal to the Euler characteristic of S^8, H P^2 or C P^4. And if M is rationally elliptic …

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single stand…

In this paper, we study the existence and non-existence result of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics. We can get some useful local …

Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed that the B-model of mirror symmetry should be described by a quantum field theory on a Calabi-Yau variety, which they called the Kodaira-Spenser theory (we call it the BCOV theory). This is the first of three papers in which we construct and analyze the quantum BCOV theory…

Modern treatment of Winger's pencil reveals deep connections to modular curves and monodromy.

The paper studies minimal resistance dynamics in radial fields, finding unique solutions for incompressible flows.

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…

We prove that the moduli space of the pseudo holomorphic curves in the A-model on a symplectic torus is homeomorphic to a moduli space of Feynman diagrams in the configuration space of the morphisms in the B-model on the corresponding elliptic curve. These moduli spaces determine the $A_{\infty}$ structure of the both …

Given a solution $u$ to a linear homogeneous second order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set $\Cr(u)\equiv \{x:|\nabla u|(x)=0\}$. The results are new even for harmonic functions on $\dR^n$. Given such a $u$, the standard {\it first o…

Generalized complex geometry, introduced by Hitchin, encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define a…

The paper connects orbifold singularities to higher symmetries in SQFTs.

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary d…

In this paper we will consider the 2-fold symmetric complex hyperbolic triangle groups generated by three complex reflections through angle 2pi/p with p no smaller than 2. We will mainly concentrate on the groups where some elements are elliptic of finite order. Then we will classify all such groups which are candidate…

The paper examines solutions to a specific type of nonlinear equation in a disk, proving existence and uniqueness.