Journal of Current Glaucoma Practice

Abstract Let G be a real noncompact semisimple connected Lie group and let $$\rho : G \longrightarrow \text {SL}(V)$$ ρ : G ⟶ SL ( V ) be a faithful irreducible representation on a finite-dimensional vector space V over $$\mathbb {R}$$ R . We suppose that there exists a scalar product $$\texttt {g}$$ g on V such that $$\rho (G)=K\exp ({\mathfrak {p}})$$ ρ ( G ) = K exp ( p ) , where $$K=\text {SO}(V,\texttt {g})\cap \rho (G)$$ K = SO ( V , g ) ∩ ρ ( G ) and $${\mathfrak {p}}=\text {Sym}_o (V,\texttt {g})\cap (\text {d} \rho )_e ({\mathfrak {g}})$$ p = Sym o ( V , g ) ∩ ( d ρ ) e ( g ) . Here, $${\mathfrak {g}}$$ g denotes the Lie algebra of G, $$\text {SO}(V,\texttt {g})$$ SO ( V , g ) denotes the connected component of the orthogonal group containing the identity element and $$\text {Sym}_o (V,\texttt {g})$$ Sym o ( V , g ) denotes the set of symmetric endomorphisms of V with trace zero. In this paper, we study the projective representation of G on $${\mathbb {P}}(V)$$ P ( V ) arising from $$\rho $$ ρ . There is a corresponding G-gradient map $$\mu _{\mathfrak {p}}:{\mathbb {P}}(V) \longrightarrow {\mathfrak {p}}$$ μ p : P ( V ) ⟶ p . Using G-gradient map techniques, we prove that the unique compact G orbit $${\mathcal {O}}$$ O inside the unique compact $$U^\mathbb {C}$$ U C orbit $${\mathcal {O}}'$$ O ′ in $${\mathbb {P}} (V^\mathbb {C})$$ P ( V C ) , where U is the semisimple connected compact Lie group with Lie algebra $${\mathfrak {k}} \oplus {\textbf {i}} {\mathfrak {p}}\subseteq \mathfrak {sl}(V^\mathbb {C})$$ k ⊕ i p ⊆ sl ( V C ) , is the set of fixed points of an anti-holomorphic involutive isometry of $${\mathcal {O}}'$$ O ′ and so a totally geodesic Lagrangian submanifold of $${\mathcal {O}}'$$ O ′ . Moreover, $${\mathcal {O}}$$ O is contained in $${\mathbb {P}}(V)$$ P ( V ) . The restriction of the function $$\mu _{\mathfrak {p}}^\beta (x):=\langle \mu _{\mathfrak {p}}(x),\beta \rangle $$ μ p β ( x ) : = ⟨ μ p ( x ) , β ⟩ , where $$\langle \cdot , \cdot \rangle $$ ⟨ · , · ⟩ is an $$\text {Ad}(K)$$ Ad ( K ) -invariant scalar product on $${\mathfrak {p}}$$ p , to $${\mathcal {O}}$$ 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 $$\mu _{\mathfrak {p}}({\mathbb {P}}(V))$$ μ p ( P ( V ) ) .

Abstract 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.

In this article we study covering spaces of symplectic toric orbifolds and symplectic toric orbifold bundles. In particular, we show that all symplectic toric orbifold coverings are quotients of some symplectic toric orbifold by a finite subgroup of a torus. We then give a general description of the labeled polytope of a toric orbifold bundle in terms of the polytopes of the fiber and the base. Finally, we apply our findings to study the number of toric structures on products of labeled projective spaces.

In this article, we study a locally constrained fully nonlinear curvature flow for convex capillary hypersurfaces in half-space. We prove that the flow preserves the convexity, exists for all time, and converges smoothly to a spherical cap. This can be viewed as the fully nonlinear counterpart of the result in Mei et al. (Int Math Res Not IMRN 1:152–174, 2024). As a byproduct, a high-order capillary isoperimetric ratio (1.6) evolves monotonically along this flow, which yields a class of the Alexandrov–Fenchel inequalities.

Abstract We study the Boothby–Wang fibration of para-Sasakian manifolds and introduce the class of para-Sasakian $$\phi $$ ϕ -symmetric spaces, canonically fibering over para-Hermitian symmetric spaces. We remark that in contrast to the Hermitian setting the center of the isotropy group of a simple para-Hermitian symmetric space G/H can be either one- or two-dimensional, and prove that the associated metric is not necessarily the G-invariant extension of the Killing form of G. Using the Boothby–Wang fibration and the classification of semisimple para-Hermitian symmetric spaces, we explicitly construct semisimple para-Sasakian $$\phi $$ ϕ -symmetric spaces fibering over semisimple para-Hermitian symmetric spaces. We provide moreover an example of non-semisimple para-Sasakian $$\phi $$ ϕ -symmetric space.

A principal torus bundle over a complex manifold with even dimensional fiber and characteristic class of type (1, 1) admits a family of regular generalized complex structures (GCS) with the fibers as leaves of the associated symplectic foliation. We show that such a generalized complex structure is equivalent to the product of the complex structure on the base and the symplectic structure on the fiber in a tubular neighborhood of an arbitrary fiber if and only if the bundle is flat. This has consequences for the generalized Dolbeault cohomology of the bundle that includes a Künneth formula. On a more general note, if a principal bundle over a complex manifold with a symplectic structure group admits a GCS with the fibers of the bundle as leaves of the associated symplectic foliation, and the GCS is equivalent to a product GCS in a neighborhood of every fiber, then the bundle is flat and symplectic.

Abstract We consider two different $$\text {SU}(2)^2$$ SU ( 2 ) 2 -invariant cohomogeneity one manifolds, one non-compact $$M=\mathbb {R}^4 \times S^3$$ M = R 4 × S 3 and one compact $$M=S^4 \times S^3$$ M = S 4 × S 3 , and study the existence of coclosed $$\text {SU}(2)^2$$ SU ( 2 ) 2 -invariant $$G_2$$ G 2 -structures constructed from half-flat $$\text {SU}(3)$$ SU ( 3 ) -structures. For $$\mathbb {R}^4 \times S^3$$ R 4 × S 3 , we prove the existence of a family of coclosed (but not necessarily torsion-free) $$G_2$$ G 2 -structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed $$G_2$$ G 2 -structure constructed from a half-flat $$\text {SU}(3)$$ SU ( 3 ) -structure is in this family. For $$S^4 \times S^3$$ S 4 × S 3 , we prove that there are no $$\text {SU}(2)^2$$ SU ( 2 ) 2 -invariant coclosed $$G_2$$ G 2 -structures constructed from half-flat $$\text {SU}(3)$$ SU ( 3 ) -structures.

Let (M, L) be a (compact) non-spin spin $$^c$$ manifold. Fix a Riemannian metric g on M and a connection A on L, and let $$D_L$$ be the associated spin $$^c$$ Dirac operator. Let $$R^{\text {tw }}_{(g,A)}:=R_g + 2ic(\Omega )$$ be the twisted scalar curvature (which takes values in the endomorphisms of the spinor bundle), where $$R_g$$ is the scalar curvature of g and $$2ic(\Omega )$$ comes from the curvature 2-form $$\Omega $$ of the connection A. Then the Lichnerowicz-Schrödinger formula for the square of the Dirac operator takes the form $$D_L^2 =\nabla ^*\nabla + \frac{1}{4}R^{\text {tw }}_{(g,A)}$$ . In a previous work we proved that a closed non-spin simply-connected spin $$^c$$ -manifold (M, L) of dimension $$n\ge 5$$ admits a pair (g, A) such that $$R^{\text {tw }}_{(g,A)}>0$$ if and only if the index $$\alpha ^c(M,L):={\text {ind}}D_L$$ vanishes in $$K_n$$ . In this paper we introduce a scalar-valued generalized scalar curvature $$R^{\text {gen }}_{(g,A)}:=R_g - 2|\Omega |_{op}$$ , where $$|\Omega |_{op}$$ is the pointwise operator norm of Clifford multiplication $$c(\Omega )$$ , acting on spinors. We show that the positivity condition on the operator $$R^{\text {tw }}_{(g,A)}$$ is equivalent to the positivity of the scalar function $$R^{\text {gen }}_{(g,A)}$$ . We prove a corresponding trichotomy theorem concerning the curvature $$R^{\text {gen }}_{(g,A)}$$ , and study its implications. We also show that the space $$\mathcal {R}^{{\textrm{gen}+}}(M,L)$$ of pairs (g, A) with $$R^{\text {gen }}_{(g,A)}>0$$ has non-trivial topology, and address a conjecture about non-triviality of the “index difference” map.

We consider the problem of finding conformal metrics on the standard half sphere with prescribed scalar curvature and zero-boundary mean curvature. We prove a perturbation result when the curvature function is flat near its boundary critical points. As a product we extend some previous well known results and provide an entirely new one.

On a compact Riemannian manifold M with boundary Y, we express the log of the zeta-determinant of the Dirichlet-to-Neumann operator acting on q-forms on Y as the difference of the log of the zeta-determinant of the Laplacian on q-forms on M with the absolute boundary condition and that of the Laplacian with the Dirichlet boundary condition with an additional term which is expressed by curvature tensors. When the dimension of M is 2 and 3, we compute these terms explicitly. We also discuss the value of the zeta function at zero associated to the Dirichlet-to-Neumann operator by using a metric rescaling method. As an application, we recover the result of the conformal invariance obtained in Guillarmou and Guillope (Int Math Res Not IMRN 2007(22):rnm099, 2007) when $${\text {dim}}M = 2$$ .

AbstractIn this paper we focus on the interplay between the behaviour of the Frölicher spectral sequence and the existence of special Hermitian metrics on the manifold, such as balanced, SKT or generalized Gauduchon. The study of balanced metrics on nilmanifolds endowed with strongly non-nilpotent complex structures allows us to provide infinite families of compact balanced manifolds with Frölicher spectral sequence not degenerating at the second page. Moreover, this result is extended to non-degeneration at any arbitrary page. Similar results are obtained for the Frölicher spectral sequence of compact generalized Gauduchon manifolds. We also find a compact SKT manifold whose Frölicher spectral sequence does not degenerate at the second page, thus providing a counterexample to a conjecture by Popovici.