Lagrangian isotopy of tori in $${S^2\times S^2}$$ and $${{\mathbb{C}}P^2}$$CP2

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space \({{\mathbb{R}}^4}\), the projective plane \({{\mathbb{C}}P^2}\), and the monotone \({S^2 \times S^2}\). The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for \({T^*{\mathbb{T}}^2}\), i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.     

On Willmore Legendrian surfaces in $$\mathbb {S}^5$$ and the contact stationary Legendrian Willmore surfaces

In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in Luo (arXiv:1211.4227v6) to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in \(\mathbb {S}^5\), and then we use this relation to prove a classification result for Willmore Legendrian spheres in \(\mathbb {S}^5\). We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface in \(\mathbb {S}^5\) belongs to [0, 2], then it must be 0 and L is totally geodesic or 2 and L is a flat minimal Legendrian tori, which generalizes the result of Yamaguchi et al. (Proc Am Math Soc 54:276–280, 1976). We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Let \(\Sigma \) be a closed surface and \((M,\alpha ,g_\alpha ,J)\) a 5-dimensional Sasakian manifold with a contact form \(\alpha \), an associated metric \(g_\alpha \) and an almost complex structure J. Assume that \(f:\Sigma \mapsto M\) is a Legendrian immersion. Then f is called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed if \((M,\alpha ,g_\alpha ,J)\) is a Sasakian Einstein manifold, in particular \(\mathbb {S}^5\).     

New Characterizations of the Clifford Torus as a Lagrangian Self-Shrinker

In this paper, we obtain several new characterizations of the Clifford torus as a Lagrangian self-shrinker. We first show that the Clifford torus \({\mathbb {S}}^1(1)\times {\mathbb {S}}^1(1)\) is the unique compact orientable Lagrangian self-shrinker in \({\mathbb {C}}^2\) with \(|A|^2\le 2\), which gives an affirmative answer to Castro–Lerma’s conjecture in Castro and Lerma (Int Math Res Not 6:1515–1527; 2014). We also prove that the Clifford torus is the unique compact orientable embedded Lagrangian self-shrinker with nonnegative or nonpositive Gauss curvature in \({\mathbb {C}}^2\).     

rthogonality Property $$\mathcal {}$$ and Compact Perturbations

A bounded linear operator T acting on a Hilbert space is said to have orthogonality property \(\mathcal {O}\) if the subspaces \(\ker (T-\alpha )\) and \(\ker (T-\beta )\) are orthogonal for all \(\alpha , \beta \in \sigma _p(T)\) with \(\alpha \ne \beta \). In this paper, the authors investigate the compact perturbations of operators with orthogonality property \(\mathcal {O}\). We give a sufficient and necessary condition to determine when an operator T has the following property: for each \(\varepsilon >0\), there exists \(K\in \mathcal {K(H)}\) with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) has orthogonality property \(\mathcal {O}\). Also, we study the stability of orthogonality property \(\mathcal {O}\) under small compact perturbations and analytic functional calculus.     

A New Approach to the Parameterization Method for Lagrangian Tori of Hamiltonian Systems

We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorov’s non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equation of the parameterization of the torus by three conditions which are altogether equivalent to invariance. We construct a quasi-Newton method by solving, approximately, the linearization of the functional equations defined by these three conditions around an approximate solution. Instead of dealing with the invariance error as a single source of error, we consider three different errors that take account of the Lagrangian character of the torus and the preservation of both energy and frequency. The condition of convergence reflects at which level contributes each of these errors to the total error of the parameterization. We do not require the system to be nearly integrable or to be written in action-angle variables. For nearly integrable Hamiltonians, the Lebesgue measure of the holes between invariant tori predicted by this parameterization result is of \({\mathcal {O}}(\varepsilon ^{1/2})\), where \(\varepsilon \) is the size of the perturbation. This estimate coincides with the one provided by the KAM theorem.     

Rotational Surfaces in $$S^{3}$$ with Constant Mean Curvature

There is a two-parametric family of rotational symmetric CMC surfaces; more precisely, for every real number H and every \(C\ge 2(H+\sqrt{1+H^2})\) there is a rotational symmetry surface \(\Sigma _{H,C}\) with mean curvature H. Perdomo (Asian J Math 14:73–108, 2010) showed that for every H between \(\cot \left( \frac{\pi }{m}\right) \) and \(\frac{m^2-2}{2\sqrt{m^2-1}}\) there exists an embedded rotational symmetric example with non-constant principal curvatures that is invariant under the cyclic group \(Z_m\). Recently Andrews and Li (J Differ Geom 99:169–189, 2015) showed that these embedded CMC tori are the only embedded genus 1 surfaces with CMC on the sphere. In this paper we complete the study of this family of CMC surfaces and we show that for every integer \(m>2\), there is a properly immersed example in this family that contains a great circle and is invariant under the cyclic group \(Z_m\). We will say that these examples contain the axis of symmetry. We also show that every non-isoparametric surface \(\Sigma _{H,C}\) is either properly immersed and invariant under the cyclic group \(Z_m\) for some integer \(m>1\) or it is dense in the region bounded by two isoparametric tori if the surface \(\Sigma _{H,C}\) does not contain the axis of symmetry or it is dense in the region bounded by a totally umbilical surface if the surface \(\Sigma _{H,C}\) contains the axis of symmetry.     

Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$

For \(\alpha , \beta \in L^{\infty } (S^1),\) the singular integral operator \(S_{\alpha ,\beta }\) on \(L^2 (S^1)\) is defined by \(S_{\alpha ,\beta }f:= \alpha Pf+\beta Qf\), where P denotes the orthogonal projection of \(L^2(S^1)\) onto the Hardy space \(H^2(S^1),\) and Q denotes the orthogonal projection onto \(H^2(S^1)^{\perp }\). In a recent paper, Nakazi and Yamamoto have studied the normality and self-adjointness of \(S_{\alpha ,\beta }\). This work has shown that \(S_{\alpha ,\beta }\) may have analogous properties to that of the Toeplitz operator. In this paper, we study several other properties of \(S_{\alpha ,\beta }\).     

The $${\square_{b}}$$ heat equation and multipliers via the wave equation

Recently, Nagel and Stein studied the \({\square_{b}}\) -heat equation, where \({\square_{b}}\) is the Kohn Laplacian on the boundary of a weakly pseudoconvex domain of finite type in \({\mathbb{C}^2}\) . They showed that the Schwartz kernel of \({e^{-t\square_{b}}}\) satisfies good “off-diagonal” estimates, while that of \({e^{-t\square_{b}}-\pi}\) satisfies good “on-diagonal” estimates, where π denotes the Szegö projection. We offer a simple proof of these results, which easily generalizes to other, similar situations. Our methods involve adapting the well-known relationship between the heat equation and the finite propagation speed of the wave equation to this situation. In addition, we apply these methods to study multipliers of the form \({m(\square_{b})}\) . In particular, we show that \({m(\square_{b})}\) is an NIS operator, where m satisfies an appropriate Mihlin–Hörmander condition.     

Equations and rational points of the modular curves $$X_0^+(p)$$

Let p be an odd prime number and let \(X_0^+(p)\) be the quotient of the classical modular curve \(X_0(p)\) by the action of the Atkin–Lehner operator \(w_p\). In this paper, we show how to compute explicit equations for the canonical model of \(X_0^+(p)\). Then we show how to compute the modular parametrization, when it exists, from \(X_0^+(p)\) to an isogeny factor E of dimension 1 of its Jacobian \(J_0^+(p)\). Finally, we show how to use this map to determine the rational points on \(X_0^+(p)\) up to a large fixed height.     

Affine ADE bundles over surfaces with $$ p _{g}=0$$

Given any Kodaira curve C in a complex surface X, we construct a simply-laced affine Lie algebra bundle \(\mathcal {E}\) over X. When \( p _{g}(X)=0\), we construct deformations of holomorphic structures on \(\mathcal {E}\) such that the new bundle is trivial over any ADE curve \( C^{\prime }\) inside C and therefore descends to the singular surface obtained by contracting \(C^{\prime }\).     

Deformation of the $$\sigma _2$$-curvature

Our main goal in this work is to deal with results concern to the \(\sigma _2\)-curvature. First we find a symmetric 2-tensor canonically associated to the \(\sigma _2\)-curvature and we present an almost-Schur-type lemma. Using this tensor, we introduce the notion of \(\sigma _2\)-singular space and under a certain hypothesis we prove a rigidity result. Also we deal with the relations between flat metrics and \(\sigma _2\)-curvature. With a suitable condition on the \(\sigma _2\)-curvature, we show that a metric has to be flat if it is close to a flat metric. We conclude this paper by proving that the three-dimensional torus does not admit a metric with constant scalar curvature and nonnegative \(\sigma _2\)-curvature unless it is flat.     

$$L^1$$-Uniqueness of Kolmogorov Operators Associated with Two-Dimensional Stochastic Navier–Stokes Coriolis Equations with Space–Time White Noise

We consider the Kolmogorov operator \(K\) associated with a stochastic Navier–Stokes equation driven by space–time white noise on the two-dimensional torus with periodic boundary conditions and a rotating reference frame, introducing fictitious forces such as the Coriolis force. This equation then serves as a simple model for geophysical flows. We prove that the Gaussian measure induced by the enstrophy is infinitesimally invariant for \(K\) on finitely based cylindrical test functions, and moreover, \(K\) is \(L^1\)-unique with respect to the enstrophy measure for sufficiently large viscosity.     

Actions of $$\mathbf {SAut}\varvec{(F_n)}$$

We study actions of SAut\((F_n)\), the unique subgroup of index two in the automorphism group of a free group of rank n, and obtain rigidity results for its representations. In particular, we show that every smooth action of SAut\((F_n)\) on a low dimensional torus is trivial.     

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\). We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in \(\mathbb {S}^3 \times \mathbb {S}^3\). We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) can only be \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).     

Kakeya-Type Sets Over Cantor Sets of Directions in $$\mathbb {R}^{d+1}$$

Given a Cantor-type subset \(\Omega \) of a smooth curve in \(\mathbb R^{d+1}\), we construct examples of sets that contain unit line segments with directions from \(\Omega \) and exhibit analytical features similar to those of classical Kakeya sets of arbitrarily small \((d+1)\)-dimensional Lebesgue measure. The construction is based on probabilistic methods relying on the tree structure of \(\Omega \), and extends to higher dimensions an analogous planar result of Bateman and Katz (Math Res Lett 15(1):73–81, 2008). In contrast to the planar situation, a significant aspect of our analysis is the classification of intersecting tube tuples relative to their location, and the deduction of intersection probabilities of such tubes generated by a random mechanism. The existence of these Kakeya-type sets implies that the directional maximal operator associated with the direction set \(\Omega \) is unbounded on \(L^p(\mathbb {R}^{d+1})\) for all \(1\le p<\infty \).     

Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian \(L:T^{(k)}Q\rightarrow {\mathbb {R}}\) with \(k\ge 1\), the resulting discrete equations define a generally implicit numerical integrator algorithm on \(T^{(k-1)}Q\times T^{(k-1)}Q\) that approximates the flow of the higher-order Euler–Lagrange equations for L. The algorithm equations are called higher-order discrete Euler–Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. We construct an exact discrete Lagrangian \(L_d^e\) using the locally unique solution of the higher-order Euler–Lagrange equations for L with boundary conditions. By taking the discrete Lagrangian as an approximation of \(L_d^e\), we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem.     

Weighted $${{L^p}}$$-Liouville theorems for hypoelliptic partial differential operators on Lie groups

We prove weighted \({L^p}\)-Liouville theorems for a class of second-order hypoelliptic partial differential operators \({\mathcal{L}}\) on Lie groups \({\mathbb{G}}\) whose underlying manifold is \({n}\)-dimensional space. We show that a natural weight is the right-invariant measure \(\check{H}\) of \({\mathbb{G}}\). We also prove Liouville-type theorems for \({C^{2}}\) subsolutions in \({L^{p}(\mathbb{G},\check{H})}\). We provide examples of operators to which our results apply, jointly with an application to the uniqueness for the Cauchy problem for the evolution operator \({\mathcal{L}-\partial_{t}}\).     

On the image,characterization, and automatic continuity of $$\varvec{(\sigma }, \varvec{\tau }$$ )-derivations

The first main theorem of this paper asserts that any \((\sigma , \tau )\)-derivation d, under certain conditions, either is a \(\sigma \)-derivation or is a scalar multiple of (\(\sigma - \tau \)), i.e. \(d = \lambda (\sigma - \tau )\) for some \(\lambda \in \mathbb {C} \backslash \{0\}\). By using this characterization, we achieve a result concerning the automatic continuity of \((\sigma , \tau \))-derivations on Banach algebras which reads as follows. Let \(\mathcal {A}\) be a unital, commutative, semi-simple Banach algebra, and let \(\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}\) be two distinct endomorphisms such that \(\varphi \sigma (\mathbf e )\) and \(\varphi \tau (\mathbf e )\) are non-zero complex numbers for all \(\varphi \in \Phi _\mathcal {A}\). If \(d : \mathcal {A} \rightarrow \mathcal {A}\) is a \((\sigma , \tau )\)-derivation such that \(\varphi d\) is a non-zero linear functional for every \(\varphi \in \Phi _\mathcal {A}\), then d is automatically continuous. As another objective of this research, we prove that if \(\mathfrak {M}\) is a commutative von Neumann algebra and \(\sigma :\mathfrak {M} \rightarrow \mathfrak {M}\) is an endomorphism, then every Jordan \(\sigma \)-derivation \(d:\mathfrak {M} \rightarrow \mathfrak {M}\) is identically zero.