Annals of Global Analysis and Geometry最新文献

Given a noncompact disconnected periodic curve (Gamma ) of infinite length with two components and no self-intersection in (mathbb R^3), it is proved that there exists a noncompact simply connected periodic minimal surface spanning (Gamma ). As an application, it is shown that for any tetrahedron T with dihedral angles (le 90^circ ), there exist four embedded minimal annuli in T, which are perpendicular to (partial T) along their boundary. It is also proved that every Platonic solid of (mathbb R^3) contains a free boundary embedded minimal surface of genus zero.

We study the moduli space of flat maximal space-like embeddings in ({mathbb {H}}^{2,2}) from various aspects. We first describe the associated Codazzi tensors to the embedding in the general setting, and then, we introduce a family of pseudo-Kähler metrics on the moduli space. We show the existence of two Hamiltonian actions with associated moment maps and use them to find a geometric global Darboux frame for any symplectic form in the above family.

We prove that (T^2)-invariant Einstein metrics with non-negative sectional curvature on a four-manifold are locally symmetric.

We show that all non-developable ruled surfaces endowed with Ricci metrics in the three-dimensional Euclidean space may be constructed using curves of constant torsion and its binormal. This allows us to give characterizations of the helicoid as the only surface of this kind that admits a parametrization with plane line of striction, and as the only with constant mean curvature.

It was shown by Allen (in: Volume above distance below, 2020) that on a closed manifold where the diameter of a sequence of Riemannian metrics is bounded, if the volume converges to the volume of a limit manifold, and the sequence of Riemannian metrics are (C^0) converging from below then one can conclude volume preserving Sormani-Wenger Intrinsic Flat convergence. The result was extended to manifolds with boundary by Allen et al. (in: Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below, 2021) by a doubling with necks procedure which produced a closed manifold and reduced the case with boundary to the case without boundary. The consequence of the doubling with necks procedure was requiring a stronger condition than necessary on the boundary. Using the estimates for the Sormani-Wenger Intrinsic Flat distance on manifolds with boundary developed by Allen et al. (in: Intrinsic flat stability of manifolds with boundary where volume converges and distance is bounded below, 2021), we show that only a bound on the area of the boundary is needed in order to conclude volume preserving intrinsic flat convergence for manifolds with boundary. We also provide an example which shows that one should not expect convergence without a bound on area.

We construct smooth symplectic resolutions of the quotient of ({mathbb {R}}^2 ) under some infinite discrete sub-group of ({textrm{ GL}}_2({mathbb {R}}) ) preserving a log-symplectic structure. This extends from algebraic geometry to smooth real differential geometry the Du Val symplectic resolution of ({mathbb {C}}^2 hspace{-1.5pt} / hspace{-1.5pt}G), with (G subset {textrm{ SL}}_2({mathbb {C}}) ) a finite group. The first of these infinite groups is (G={mathbb {Z}}), identified to triangular matrices with spectrum ({1} ). Smooth functions on the quotient (mathbb {R}^2 hspace{-1.5pt} / hspace{-1.5pt} G ) come with a natural Poisson bracket, and (mathbb {R}^2hspace{-1.5pt} / hspace{-1.5pt}G) is for an arbitrary (k ge 1) set-isomorphic to the real Du Val singular variety (A_{2k} = {(x,y,z) in {mathbb {R}}^3, x^2 +y^2= z^{2k}}). We show that each one of the usual minimal resolutions of these Du Val varieties are symplectic resolutions of (mathbb {R}^2hspace{-1.5pt} / hspace{-1.5pt}G). The same holds for (G'={mathbb {Z}} rtimes {mathbb {Z}}hspace{-1.5pt} / hspace{-1.5pt}2mathbb {Z}) (identified to triangular matrices with spectrum ({pm 1} )), with the upper half of the Du Val singularity (D_{2k+1} ) playing the role of (A_{2k}).

Let (M, J) be a 2n-dimensional almost complex manifold and let (xin M). We define the notion of almost complex blow-up of (M, J) at x. We prove the existence of almost complex blow-ups at x under suitable assumptions on the almost complex structure J and we provide explicit examples of such a construction. We note that almost complex blow-ups are unique if they exist. When (M, J) is a 4-dimensional almost complex manifold, we give an obstruction on J to the existence of almost complex blow-ups at a point and prove that the almost complex blow-up at a point of a compact almost Kähler manifold is almost Kähler.

We obtain a correspondence between irreducible real parallel spinors on pseudo-Riemannian manifolds (M, g) of signature (4, 3) and solutions of an associated differential system for three-forms that satisfy a homogeneous algebraic equation of order two in the Kähler-Atiyah bundle of (M, g). Applying this general framework, we obtain an intrinsic algebraic characterization of (text {G}_2^*)-structures as well as the first explicit description of isotropic irreducible spinors in signature (4, 3) that are parallel under a general connection on the spinor bundle. This description is given in terms of a coherent system of mutually orthogonal and isotropic one-forms and follows from the characterization of the stabilizer of an isotropic spinor as the stabilizer of a highly degenerate three-form that we construct explicitly. Using this result, we show that isotropic spinors parallel under a metric connection with torsion exist when the connection preserves the aforementioned coherent system. This allows us to construct a natural class of metrics of signature (4, 3) on (mathbb {R}^7) that admit spinors parallel under a metric connection with torsion.

Let G be a real noncompact semisimple connected Lie group and let (rho : G longrightarrow text {SL}(V)) be a faithful irreducible representation on a finite-dimensional vector space V over (mathbb {R}). We suppose that there exists a scalar product (texttt {g}) on V such that (rho (G)=Kexp ({mathfrak {p}})), where (K=text {SO}(V,texttt {g})cap rho (G)) and ({mathfrak {p}}=text {Sym}_o (V,texttt {g})cap (text {d} rho )_e ({mathfrak {g}})). Here, ({mathfrak {g}}) denotes the Lie algebra of G, (text {SO}(V,texttt {g})) denotes the connected component of the orthogonal group containing the identity element and (text {Sym}_o (V,texttt {g})) denotes the set of symmetric endomorphisms of V with trace zero. In this paper, we study the projective representation of G on ({mathbb {P}}(V)) arising from (rho ). There is a corresponding G-gradient map (mu _{mathfrak {p}}:{mathbb {P}}(V) longrightarrow {mathfrak {p}}). Using G-gradient map techniques, we prove that the unique compact G orbit ({mathcal {O}}) inside the unique compact (U^mathbb {C}) orbit ({mathcal {O}}') in ({mathbb {P}} (V^mathbb {C})), where U is the semisimple connected compact Lie group with Lie algebra ({mathfrak {k}} oplus {textbf {i}} {mathfrak {p}}subseteq mathfrak {sl}(V^mathbb {C})), is the set of fixed points of an anti-holomorphic involutive isometry of ({mathcal {O}}') and so a totally geodesic Lagrangian submanifold of ({mathcal {O}}'). Moreover, ({mathcal {O}}) is contained in ({mathbb {P}}(V)). The restriction of the function (mu _{mathfrak {p}}^beta (x):=langle mu _{mathfrak {p}}(x),beta rangle ), where (langle cdot , cdot rangle ) is an (text {Ad}(K))-invariant scalar product on ({mathfrak {p}}), to ({mathcal {O}}) achieves the maximum on the unique compact orbit of a suitable parabolic subgroup and this orbit is connected. We also describe the irreducible representations of parabolic subgroups of G in terms of the facial structure of the convex body given by the convex envelope of the image (mu _{mathfrak {p}}({mathbb {P}}(V))).

We demonstrate the existence and uniqueness of the solution to the Dirichlet problem for a generalization of Hitchin’s equation for diagonal harmonic metrics on cyclic Higgs bundles. The generalized equations are formulated using subharmonic functions. In this generalization, the coefficient exhibits worse regularity than that in the original equation.