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Register a new userA new approach to the Fraser-Li conjecture with the Weierstrass representation formula
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In this paper, we provide a sufficient condition for a curve on a surface in $mathbb{R}^3$ to be given by an orthogonal intersection with a sphere. This result makes it possible to express the boundary condition entirely in terms of the Weierstrass data without integration when dealing with free boundary minimal surfaces in a ball $mathbb{B}^3$. Moreover, we show that the Gauss map of an embedded free boundary minimal annulus is one to one. By using this, the Fraser-Li conjecture can be translated into the problem of determining the Gauss map. On the other hand, we show that the Liouville type boundary value problem in an annulus gives some new insight into the structure of immersed minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic approach to the Fraser-Li conjecture.
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The Han-Li conjecture states that: Let $(M,g_0)$ be an $n$-dimensional $(ngeq 3)$ smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and $c$ be any real number, then there exists a conformal metric of $g_0$ with scalar curvature $1$ and boundary mean curvature $c$. Combining with Z. C. Han and Y. Y. Lis results, we answer this conjecture affirmatively except for the case that $ngeq 8$, the boundary is umbilic, the Weyl tensor of $M$ vanishes on the boundary and has a non-zero interior point.
Lehmann, Symanzik and Zimmermann (LSZ) proved a theorem showing how to obtain the S-matrix from time-ordered Green functions. Their result, the reduction formula, is fundamental to practical calculations of scattering processes. A known problem is that the operators that they use to create asymptotic states create much else besides the intended particles for a scattering process. In the infinite-time limits appropriate to scattering, the extra contributions only disappear in matrix elements with normalizable states, rather than in the created states themselves, i.e., the infinite-time limits of the LSZ creation operators are weak limits. The extra particles that are created are in a different region of space-time than the intended scattering process. To be able to work with particle creation at non-asymptotic times, e.g., to give a transparent and fully deductive treatment for scattering with long-lived unstable particles, it is necessary to have operators for which the infinite-time limits are strong limits. In this paper, I give an improved method of constructing such operators. I use them to give an improved systematic account of scattering theory in relativistic quantum field theories, including a new proof of the reduction formula. I make explicit calculations to illustrate the problems with the LSZ operators and their solution with the new operators. Not only do these verify the existence of the extra particles created by the LSZ operators and indicate a physical interpretation, but they also show that the extra components are so large that their contribution to the norm of the state is ultra-violet divergent in renormalizable theories. Finally, I discuss the relation of this work to the work of Haag and Ruelle on scattering theory.
We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang.
Given any admissible $k$-dimensional family of immersions of a given closed oriented surface into an arbitrary closed Riemannian manifold, we prove that the corresponding min-max width for the area is achieved by a smooth (possibly branched) immersed minimal surface with multiplicity one and Morse index bounded by $k$.
We present a representation formula for discrete indefinite affine spheres via loop group factorizations. This formula is derived from the Birkhoff decomposition of loop groups associated with discrete indefinite affine spheres. In particular we show that a discrete indefinite improper affine sphere can be constructed from two discrete plane curves.
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