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We prove an analog of Schoen--Yau univalentness theorem for saddle maps between discs.
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We show that the connectedness of the set of parameters for which the over-rotation interval of a bimodal interval map is constant. In other words, the over-rotation interval is a monotone function of a bimodal interval map.
We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients. This is done by ruling out the existence of certain post-critical curves in the moduli space of aforementioned maps, and confirms a conjecture made in [Fil19] based on experimental evidence.
Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $Omega$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geometric isoperimetric inequalities under finiteness of the positive part of total Gauss curvature of a conformal metric on $Omega$. Consequently, new analytic and geometric isoperimetric-type inequalities are discovered. Furthermore, when applying the geometric principle to two-dimensional Riemannian manifolds, we find fortunately that ${0,1}$-form of the induced principle is midway between Moser-Trudingers inequality and Nash-Sobolevs inequality on complete noncompact boundary-free surfaces, and yet equivalent to Nash-Sobolevs/Faber-Krahns eigenvalue/Heat-kernel-upper-bound/Log-Sobolevs inequality on the surfaces with finite total Gauss curvature and quadratic area growth.
Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. Kahler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae under suitable growth conditions on the energy of the p-forms. As an application, we establish a monotonicity formula for the Ricci form of a Kahler manifold of constant scalar curvature and then get a growth condition to derive the Ricci flatness of the Kahler manifold. In particular, when the curvature does not change sign, the Kahler manifold is isometrically biholomorphic to C^m. Another application is to deduce the monotonicity formulae for volumes of minimal submanifolds in some outer spaces with suitable exhaustion functions. In this way, we recapture the classical volume monotonicity formulae of minimal submanifolds in Euclidean spaces. We also apply the vanishing theorems to Bernstein type problem of submanifolds in Euclidean spaces with parallel mean curvature. In particular, we may obtain Bernstein type results for minimal submanifolds, especially for minimal real Kahler submanifolds under weaker conditions.
In 1939 H. Weyl has introduced the so called intrinsic volumes $V_i(M^n), i=0,dots,n$, (known also as Lipschitz-Killing curvatures) for any closed smooth Riemannian manifold $M^n$. Given a Riemmanian submersion of compact smooth Riemannian manifolds $Mto B$, $B$ is connected. For $varepsilon >0$ let us define a new Riemannian metric on $M$ by multiplying the original one by $varepsilon$ along the vertical directions and keeping it the same along the (orthogonal) horizontal directions. Denote the corresponding Riemannian manifold by $M_varepsilon$. The main result says that $lim_{varepsilonto +0} V_i(M_varepsilon)=chi(Z) V_i(B)$, where $chi(Z)$ is the Euler characteristic of a fiber of the submersion. This result is consistent with more general open conjectures on convergence of intrinsic volumes formulated previously by the author.
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