In this article we apply a Bochner type formula to show that on a compact conformally flat riemannian manifold (or half-conformally flat in dimension 4) certain types of orthogonal almost-complex structures, if they exist, give the absolute minimum for the energy functional. We give a few examples when such minimizers exist, and in particular, we prove that the standard almost-complex structure on the round S^6 gives the absolute minimum for the energy. We also discuss the uniqueness of this minimum and the extension of these results to other orthogonal G-structures.
Let $(G,g)$ be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation $\F$ with minimal leaves. Let $J$ be an almost Hermitian structure on $G$ adapted to the foliation $\F$. We classify such structures $J$ which are almost Kähler $(\A\K)$, integrable $(\I)$ or Kähler $(\K)$. Hereby we construct several new multi-dimensional examples in each class.
We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called {\it contact stationary legendrian immersions}) in the Heisenberg Group ${\mathbb H}^2$ . From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. This last result happens to be in particular a central ingredient in the analysis part of the proof of the Willmore conjecture given by the author in arXiv:2007.05467
Let $(G,g)$ be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation $\mathcal{F}$ with minimal leaves. Let $J$ be an almost Hermitian structure on $G$ adapted to the foliation $\mathcal{F}$. The corresponding Lie algebra $\mathfrak{g}$ must then belong to one of 20 families $\mathfrak{g}_1,\dots,\mathfrak{g}_{20}$ according to S. Gudmundsson and M. Svensson. We classify such structures $J$ which are almost Kähler $(\mathcal{A}\mathcal{K})$, integrable $(\mathcal{I})$ or Kähler $(\mathcal{K})$. Hereby, we construct 16 multi-dimensional almost Kähler fami…
We study and classify almost complex totally geodesic submanifolds of the nearly Kaehler flag manifold $F_{1,2}(\mathbb C^3)$, and of its semi-Riemannian counterpart. We also develop a structural approach to the nearly Kaehler flag manifold $F_{1,2}(\mathbb C^3)$, expressing for example the curvature tensor in terms of the nearly Kaehler structure $J$ and the three canonical orthogonal complex structures.
We show that the only rational homology spheres which can admit almost complex structures occur in dimensions two and six. Moreover, we provide infinitely many examples of six-dimensional rational homology spheres which admit almost complex structures, and infinitely many which do not. We then show that if a closed almost complex manifold has sum of Betti numbers three, then its dimension must be a power of two.
We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
We construct a branched center manifold in a neighborhood of a singular point of a $2$-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensional integral currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.
We analyse the asymptotic behaviour of solutions of the Teichmüller harmonic map flow from cylinders, and more generally of `almost minimal cylinders', in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition forces the domain to stretch out as a boundary bubble forms. Our main result then establishes that for prescribed boundary curves that satisfy Douglas' separation condition, these boundary bubbles will not only be harmonic but will themselves be branched minimal immersions. Toget…
Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we obtain sufficient geometric conditions for the minimal surfaces spanned by a given boundary to represent all the possible limits of sequences of almost-minimal surfaces. Finally, we provide some sharp quantitative estimates on the distance of an almost-minimal surface from its limit minimal surface.
Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type $D_m$. Based on this geometric interpretation he conjectured that these polynomials all have pairwise different real roots in the interval $[\,0,1\,]$. In this article the polynomials constructed by Reiswich will be identified as special cases of Jacobi polynomials thus proving the conjecture about minimal isotropy orbits of compact symmetric spaces with Dynkin…
若Gamma是实秩大于1的半单李群中的一个非一致不可约格，我们证明了Gamma包含一个子群同构于SL（3，R），SL（3，C），或直积SL（2，R）^m x SL（2，C）^n$中的一个非一致不可约格，其中m + n > 1. (In几何术语，这可以解释为关于有限体积的、非紧的、局部对称的高阶空间的全测地子空间的存在性的陈述。结果的另一种表述是，如果G是Q（有理数）上的任何迷向的几乎单代数群，使得G的实秩大于1，则G包含迷向的几乎单Q-子群H，使得H是拟分裂的，并且H的实秩大于1。
