Let $M$ be an $n$-dimensional complete Riemannian manifold with Ricci curvature $ge n-1$. In cite{colding1, colding2}, Tobias Colding, by developing some new techniques, proved that the following three condtions: 1) $d_{GH}(M, S^n)to 0$; 2) the volum
e of $M$ ${text{Vol}}(M)to{text{Vol}}(S^n)$; 3) the radius of $M$ ${text{rad}}(M)topi$ are equivalent. In cite{peter}, Peter Petersen, by developing a different technique, gave the 4-th equivalent condition, namely he proved that the $n+1$-th eigenvalue of $M$ $lambda_{n+1}(M)to n$ is also equivalent to the radius of $M$ ${text{rad}}(M)topi$, and hence the other two. In this note, we give a new proof of Petersens theorem by utilizing Coldings techniques.
In this paper, we consider a discrete restriction associated with KdV equations. Some new Strichartz estimates are obtained. We also establish the local well-posedness for the periodic generalized Korteweg-de Vries equation with nonlinear term $ F(u)
p_x u$ provided $Fin C^5$ and the initial data $phiin H^s$ with $s>1/2$.
In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgains level set result on Strichartz estimates associated with Schrodinger equations on torus. Some sharp estimates on $L^{frac{2(d+2)}{d}}$ norm
of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on $L^2(mathbb Z)$.
We use high-resolution {sl Hubble Space Telescope} imaging observations of the young ($sim 15-25$ Myr-old) star cluster NGC 1818 in the Large Magellanic Cloud to derive an estimate for the binary fraction of F stars ($1.3 < M_star/M_odot < 1.6$). Thi
s study provides the strongest constraints yet on the binary fraction in a young star cluster in a low-metallicity environment (${[Fe/H]} sim -0.4$ dex). Employing artificial-star tests, we develop a simple method that can efficiently measure the probabilities of stellar blends and superpositions from the observed stellar catalog. We create synthetic color-magnitude diagrams matching the fundamental parameters of NGC 1818, with different binary fractions and mass-ratio distributions. We find that this method is sensitive to binaries with mass ratios, $q ga 0.4$. For binaries with F-star primaries and mass ratios $q > 0.4$, the binary fraction is $sim 0.35$. This suggests a total binary fraction for F stars of 0.55 to unity, depending on assumptions about the form of the mass-ratio distribution at low $q$.
In this note, we propose an approach to the study of the analogue for unipotent harmonic bundles of Schmids Nilpotent Orbit Theorem. Using this approach, we construct harmonic metrics on unipotent bundles over quasi-compact Kahler manifolds with care
fully controlled asymptotics near the compactifying divisor; such a metric is unique up to some isometry. Such an asymptotic behavior is canonical in some sense.
In this note, we survey our recent work concerning cohomologies of harmonic bundles on quasi-compact Kaehler manifolds.
