This paper develops a construction of families of U(1)n-2 -invariant special Lagrangian n-folds in Cn , extending the analytic framework introduced by Joyce ( n=3 ) to arbitrary dimension. By reducing the special Lagrangian condition to a quasilinear elliptic system of two-dimensional non-linear Cauchy-Riemann equations, we analyse both the resulting geometry and its degenerations at singular points. We show that the structure and multiplicity of singularities are governed by an associated polynomial arising from the symmetry reduction. Explicit examples are constructed, including affine and perturbative solutions, and are compared with the classical Harvey-Lawson U(1)n-1 -invariant submanifolds. We further show that the key elements of Joyce's analysis in the non-singular case, in particular the potential formulation and Dirichlet problem, extend to this higher-dimensional setting, with the proofs unchanged.
Lin-Yuan introduced the concept of the J-Einstein metric, which is a generalization of Bach-flat metric on 4-dimensional manifolds. In this article, we study the J-Einstein metrics as the critical point of quadratic curvature functionals involving the Ricci curvature and scalar curvature. We get some interesting results, which can be viewed as the improvement of Sheng-Wang’s conclusions on Bach-flat critical metrics (Bach-flat critical metrics for quadratic curvature functionals, Ann. Glob. Anal. Geom. 2018).
We study harmonic maps f: (M,g)→ (M̅,g̅) satisfying the curvature inequality Ric_g ≥ f^*Ric_g̅ . Using the Bochner identity, we show that if the sectional curvature of the target manifold is nonnegative along the image of the map, then every harmonic map from a compact manifold is totally geodesic, and it is constant whenever the inequality is strict at some point. Combining this result with Perelman’s proof of the Soul Conjecture, we obtain a rigidity theorem for harmonic maps into complete noncompact manifolds with nonnegative sectional curvature. In particular, if the image of the map contains a point at which all sectional curvatures of the target are strictly positive, then the soul of the tangent is a point. We also analyze the equality case of the Bochner identity and show that the images of eigendirections corresponding to eigenvalues of f^g̅ span two-planes of zero sectional curvature in the tanget manifold.
We show that the systolic area of every nonpositively curved closed surface M other than the torus is at least 1, with equality if and only if M is isometric to a square flat Klein bottle. The proof focuses on the Klein-double 4ℝℙ^2 and exploits Weil’s isoperimetric inequality and a comparison theorem involving a new kind of exponential-type map.
We investigate complex geometric properties of compact complex solvmanifolds endowed with left-invariant complex structures, focusing on the case where these manifolds admit a complex parallelizable solvmanifold as a finite covering. While our previous paper examined differences among complex solvmanifolds admitting a common complex parallelizable solvmanifold as a finite covering, the present paper is devoted to the study of their common geometric properties.
