Related papers: Variational theory for the resonant T-curvature eq…

We propose an alternative variational principle whose critical point is the algebraic plane curve associated to a matrix model (the spectral curve, i.e. the large $N$ limit of the resolvent). More generally, we consider a variational…

We formulate and prove compensated compactness theorems concerning the limiting behaviour of wedge products of weakly convergent differential forms on closed Riemannian manifolds \`{a} la Robbin--Rogers--Temple [Trans. Amer. Math. Soc. 303…

In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal…

We consider the Gelfand problem with rapidly growing nonlinearities in the two-dimensional bounded strictly convex domains. In this paper, we prove the uniformly boundedness of finite Morse index solutions. As a result, we show that there…

We use reflecting Brownian motion (RBM) to prove the well known Gauss-Bonnet-Chern theorem for a compact Riemannian manifold with boundary. The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary…

We use the method of higher order linearization to study an inverse boundary value problem for the minimal surface equation on a Riemannian manifold $(\mathbb{R}^n,g)$, where the metric $g$ is conformally Euclidean. In particular we show…

On symplectic manifolds, we introduce a Morse-type complex with elements generated by pairs of critical points of a Morse function. The differential of the complex consists of gradient flows and an integration of the symplectic structure…

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main…

The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied.…

We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate…

We consider infinite random casual Lorentzian triangulations emerging in quantum gravity for critical values of parameters. With each vertex of the triangulation we associate a Hilbert space representing a bosonic particle moving in…

We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show…

This note is an addendum to 'Critical values of random analytic functions on complex manifolds, Indiana Univ. Math. J. 63 No. 3 (2014), 651-686.' by R.Feng and S. Zelditch (arXiv:1212.4762). In this note, we give the formula of the limiting…

The classical Morse index theorem establishes a fundamental connection between the Morse index-the number of negative eigenvalues that characterize key spectral properties of linear self-adjoint differential operators-and the count of…

Morse index theory provides an elegant and useful tool for describing several aspects of a Lagrangian system in terms of its variational properties. In the classical framework it provides an equality between the spectral properties of a…

We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive…

We examine the equilibrium conditions of a curve in space when a local energy penalty is associated with its extrinsic geometrical state characterized by its curvature and torsion. To do this we tailor the theory of deformations to the…

We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten invariants of a closed 3-manifold X with b_1 > 0 to an invariant that `counts' gradient flow lines--including closed orbits--of a circle-valued Morse function on the…

We show, using a direct variational approach, that the second boundary value problem for the Monge-Amp\`ere equation in R^n with exponential non-linearity and target a convex body P is solvable iff 0 is the barycenter of P. Combined with…

This work contains an exposition of foundations of the variational calculus in fibered manifolds. The emphasis is laid on the geometric aspects of the theory. Especially functionals defined by real functions (Lagrange functions) or…