arXiv research

Unique Kähler-Ricci solitons found for S^1-invariant metrics.

We explicitly classify all $S^1$-invariant free boundary minimal annuli and Möbius bands in $\mathbb{B}^n$. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $S^1$-invariant metrics on the annulus and Möbius band. First, we determine the supremum of the $k$-th normaliz…

A theorem of J. Hersch (1970) states that for any smooth metric on $S^2$, with total area equal to $4π$, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the standard round metric). For metrics invariant under the standard $S^1$-action on $S^2…

In this paper we show that for a generalized Berger metric $\hat{g}$ on $S^3$ close to the round metric, the conformally compact Einstein (CCE) manifold $(M, g)$ with $(S^3, [\hat{g}])$ as its conformal infinity is unique up to isometries. For the high-dimensional case, we show that if $\hat{g}$ is an $\text{SU}(k+1)$-…

We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.

Constructs instantons on a specific G_2-manifold limit.

In this paper we show that for an $\text{Sp}(k+1)$ invariant metric $\hat{g}$ on $\mathbb{S}^{4k+3}$ $(k\geq 1)$ close to the round metric, the conformally compact Einstein (CCE) manifold $(M, g)$ with $(\mathbb{S}^{4k+3}, [\hat{g}])$ as its conformal infinity is unique up to isometries. Moreover, by the result in [LiQ…

Computer-assisted method finds new Einstein metrics on spheres.

In the special case of S^1 invariant metrics on S^2, we find necessary and sufficient conditions for the existence of isometric embeddings into the canonical R^3, in other words: a Weyl type theorem with converse.

Study classifies Kähler-Einstein metrics with rotational symmetries.

Complete constant positive scalar curvature metrics on S^n - {p_1, ..., p_k} admit a definite asymptotic structure; i.e. the metric is asymptotic to a specific S^{n-1}-invariant metric near the puncture points. This allows one to glue together two such metrics near their puncture points, provided the asymptotic structu…

The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly approximated by geodesics in the finite dimensional spaces of Bergman metrics. W…

Invariant complex structures on the homogeneous manifold $U(n+1)/U(n)\times U(p+1)/U(p)$ are reseached. The critical point of the functional of the scalar curvature is found.

Let $M$ be a compact orientable Seifered fibered 3-manifold without a boundary, and $α$ an $S^1$-invariant contact form on $M$. In a suitable adapted Riemannian metric to $α$, we provide a bound for the volume $\text{Vol}(M)$ and the curvature, which implies the universal tightness of the contact structure $ξ=\kerα$.

The study explores $S^1$-invariant Laplacian flow on 6-manifolds.

An upper bound on the first S^1 invariant eigenvalue of the Laplacian for invariant metrics on the 2-sphere is used to find obstructions to the existence of isometric embeddings of such metrics in (R^3,can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the surface of rev…

We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metri…

The paper finds explicit instantons on a specific 6-manifold.

In this paper, using the framework of equivariant differential geometry, we study proper $SO(p+1) \times SO(q+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+q+2$) and proper $SO(p+1)$-invariant biconservative hypersurfaces into the Euclidean space ${\mathbb R}^n$ ($n=p+2$). Mo…

The study extends knot theory to knotoids using two approaches.

The conformal infinity of a quaternionic-Kahler metric on a 4n-manifold with boundary is a codimension 3-distribution on the boundary called quaternionic contact. In dimensions 4n-1 greater than 7, a quaternionic contact structure is always the conformal infinity of a quaternionic-Kahler metric. On the contrary, in dim…

We show an equivariant bordism principle for constructing metrics of positive scalar curvature that are invariant under a given group action. Furthermore, we develop a new codimension-2 surgery technique which removes singular strata from fixed point free $S^1$-manifolds while preserving equivariant positive scalar cur…

In this paper, we consider half-flat $SU(3)$-structures and the subclasses of coupled and double structures. In the general case we show that the intrinsic torsion form $w_1^-$ is constant in each of the two subclasses. We then consider the problem of finding half-flat structures inducing Einstein metrics on homogeneou…

We give a complete description of all order 1 invariants of planar curves.

Study on Ricci flow of invariant metrics on spheres, finding new ancient solutions.

In the paper "Einstein metrics on compact simple Lie groups attached to standard triples", the authors introduced the definition of standard triples and proved that every compact simple Lie group $G$ attached to a standard triple $(G,K,H)$ admits a left-invariant Einstein metric which is not naturally reductive except …

This is the third in a series of three papers math.DG/0111324, math.DG/0111326 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using analytic methods. The three papers are surveyed in math.DG/0206016. Let…

Let ${\cal F}_λ$ be the space of tensor densities on ${\bf R}^n$ of degree $λ$ (or, equivalently, of conformal densities of degree $-λn$) considered as a module over the Lie algebra $so(p+1,q+1)$. We classify $so(p+1,q+1)$-invariant bilinear differential operators from ${\cal F}_λ\otimes{\cal F}_μ$ to~${\cal F}_ν$. The…

This is the first of three papers math.DG/0111326, math.DG/0204343 studying special Lagrangian 3-submanifolds (SL 3-folds) N in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, using analytic methods. The three papers are surveyed in math.DG/0206016. Let N be such a…

We consider the $\mathfrak{aff}(n|1)-$module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify $\mathfrak{aff}(n|1)-$invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first $\math…

The LeBrun-Mason twistor correspondences for $S^1$-invariant self-dual Zollfrei metrics are explicitly established. We give explicit formulas for the general solutions of the wave equation and the monopole equation on the de Sitter three-space under the assumption for the tameness at infinity by using Radon-type integr…

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.

This paper studies spectral properties of spheres with one equator.

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere $S^2$ endowed with $S^1$-invariant metrics, we consider the subsequence $λ_k^G$ of the spectrum of a Riemannian manifold $M$ which corresponds…

We introduce different bases for the vector space of $\mathrm{Sp}(2)\mathrm{Sp}(1)$-invariant, translation invariant continuous valuations on the quaternionic plane and determine a complete set of kinematic formulas.

We give a complete description of all order 1 invariants of spherical curves. We also identify the subspaces of all J-invariants and S-invariants, and present two equalities satisfied by any spherical curve.

In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) $k$-th Steklov eigenvalue on the disk is not maximized for a smooth metric on the disk for $k\geq 3$. For $k=1$ the classical…

New Kähler solitons found that are not $U(n)$-invariant.

We prove that the Calabi-Yau equation on the Kodaira-Thurston manifold has a unique solution for every $S^1$-invariant initial datum.

We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$-invariant metrics of positive scalar curvature on every $S^1$-manifold which has a fixed point component of codimension 2. As a consequence we can prove that there are non-invariant metr…

We study the Kähler-Ricci flow on a class of projective bundles $\mathbb{P}(\mathcal{O}_Σ\oplus L)$ over compact Kähler-Einstein manifold $Σ^n$. Assuming the initial Kähler metric $ω_0$ admits a U(1)-invariant momentum profile, we give a criterion, characterized by the triple $(Σ, L, [ω_0])$, under which the $\mathbb{P…

The spaces of Sp(n)-, Sp(n)U(1)- and Sp(n)Sp(1)- invariant, translation invariant, continuous convex valuations on the quaternionic vector space H^n are studied. Combinatorial dimension formulas involving Young diagrams and Schur polynomials are proved.

We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show …

Some elementary considerations are presented concerning Catenoids and their stability, separable minimal hypersurfaces, minimal surfaces obtainable by rotating shapes, determinantal varieties, minimal tori in S3, the minimality in Rnk of the ordered set of k orthogonal equal-length n-vectors, and U(1)-invariant minimal…

We prove that the space of gauge equivalence classes of U(1)-invariant connections on some SU(2)-principle bundles over the 4-sphere S^4 is weakly homotopy equivalent to a component of the second loop space of the 2-sphere S^2.

The paper derives Pizzetti formulae and inverts the Radon transform on spheres.

The paper studies critical points and flows of a $G_2$-Hilbert functional on manifolds with circle actions.

Two examples of $\mathrm{Diff}^+S^1$-invariant closed two-forms obtained from forms on jet bundles, which does not admit equivariant moment maps are presented. The corresponding cohomological obstruction is computed and shown to coincide with a nontrivial Lie algebra cohomology class on $H^2(\mathfrak{X}(S^1))$.