Publications mathématiques de l'IHÉS最新文献

We study the Fröhlich polaron model in (mathbf {R}^{3}), and prove a lower bound on its ground state energy as a function of the total momentum. The bound is asymptotically sharp at large coupling. In combination with a corresponding upper bound proved earlier (Mitrouskas et al. in Forum Math. Sigma 11:1–52, 2023), it shows that the energy is approximately parabolic below the continuum threshold, and that the polaron’s effective mass (defined as the semi-latus rectum of the parabola) is given by the celebrated Landau–Pekar formula. In particular, it diverges as (alpha ^{4}) for large coupling constant (alpha ).

For both the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set (mathbf {M}_{N}) of pure (N)-soliton states, and their associated multisoliton solutions. We prove that (i) the set (mathbf {M}_{N}) is a uniformly smooth manifold, and (ii) the (mathbf {M}_{N}) states are uniformly stable in (H^{s}), for each (s>-frac{1}{2}).

One main tool in our analysis is an iterated Bäcklund transform, which allows us to nonlinearly add a multisoliton to an existing soliton free state (the soliton addition map) or alternatively to remove a multisoliton from a multisoliton state (the soliton removal map). The properties and the regularity of these maps are extensively studied.

Abstract

This paper is the sequel to (Colin et al. in Publ. Math. Inst. Hautes Études Sci., 2024), and is devoted to proving some of the technical parts of the HF=ECH isomorphism.

Given an open book decomposition ((S,mathfrak{h} )) adapted to a closed, oriented 3-manifold (M), we define a chain map (Phi ) from a certain Heegaard Floer chain complex associated to ((S,mathfrak{h} )) to a certain embedded contact homology chain complex associated to ((S,mathfrak{h} )), as defined in (Colin et al. in Geom. Topol., 2024), and prove that it induces an isomorphism on the level of homology. This implies the isomorphism between the hat version of Heegaard Floer homology of (-M) and the hat version of embedded contact homology of (M).

Given a closed oriented 3-manifold (M), we establish an isomorphism between the Heegaard Floer homology group (HF^{+} (-M)) and the embedded contact homology group (ECH(M)). Starting from an open book decomposition ((S,mathfrak{h} )) of (M), we construct a chain map (Phi ^{+}) from a Heegaard Floer chain complex associated to ((S,mathfrak{h} )) to an embedded contact homology chain complex for a contact form supported by ((S,mathfrak{h} )). The chain map (Phi ^{+}) commutes up to homotopy with the (U)-maps defined on both sides and reduces to the quasi-isomorphism (Phi ) from (Colin et al. in Publ. Math. Inst. Hautes Études Sci., 2024a, 2024b) on subcomplexes defining the hat versions. Algebraic considerations then imply that the map (Phi ^{+}) is a quasi-isomorphism.

We consider integral area-minimizing 2-dimensional currents (T) in (Usubset mathbf {R}^{2+n}) with (partial T = Qleft [!![{Gamma }right ]!!]), where (Qin mathbf {N} setminus {0}) and (Gamma ) is sufficiently smooth. We prove that, if (qin Gamma ) is a point where the density of (T) is strictly below (frac{Q+1}{2}), then the current is regular at (q). The regularity is understood in the following sense: there is a neighborhood of (q) in which (T) consists of a finite number of regular minimal submanifolds meeting transversally at (Gamma ) (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for (Q=1). As a corollary, if (Omega subset mathbf {R}^{2+n}) is a bounded uniformly convex set and (Gamma subset partial Omega ) a smooth 1-dimensional closed submanifold, then any area-minimizing current (T) with (partial T = Q left [!![{Gamma }right ]!!]) is regular in a neighborhood of (Gamma ).

Abstract

Suppose that ℳ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in (mathbf {C} ^{n}) . We show that if ℳ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then ℳ is a translator. In particular in (mathbf {C} ^{2}) , all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators.