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Register a new userComparison between Second Variation of Area and Second Variation of Energy of a Minimal Surface
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The conformal parameterisation of a minimal surface is harmonic. Therefore, a minimal surface is a critical point of both the energy functional and the area functional. In this paper, we compare the Morse index of a minimal surface as a critical point of the area functional with its Morse index as a critical point of the energy functional. The difference between these indices is at most the real dimension of Teichmuller space. This comparison allows us to obtain surprisingly good upper bounds on the index of minimal surfaces of finite total curvature in Euclidean space of any dimension. We also bound the index of a minimal surface in an arbitrary Riemannian manifold by the area and genus of the surface, and the dimension and geometry of the ambient manifold.
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In this work it is shown that every integral varifold in an open subset of Euclidian space of locally bounded first variation can be covered by a countable collection of submanifolds of class C^2. Moreover, the mean curvature of each member of the collection agrees with the mean curvature of the varifold almost everywhere with respect to the varifold.
We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichmuller space. We then use this formula to determine the asymptotic behavior as $text{Re} (s) to infty$ of the second variation. As a consequence, for $m in mathbb{N}$, we obtain the complete expansion in $m$ of the curvature of the vector bundle $H^0(X_t, mathcal K_t)to tin mathcal T$ of holomorphic m-differentials over the Teichmuller space $mathcal T$, for $m$ large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $O(m^2 e^{-l_0 m}),$ where $l_0$ is the length of the shortest closed hyperbolic geodesic.
The emergence of the magnetic field through the photosphere has multiple manifestations and sunspots are the most prominent examples of this. One of the most relevant sunspot properties, to study both its structure and evolution, is the sunspot area: either total, umbra or penumbra area. Recently Schlichenmaier et al. (2010) studied the evolution of the active region (AR) NOAA 11024 concluding that during the penumbra formation the umbra area remains constant and that the increase of the total sunspot area is caused exclusively by the penumbra growth. In this presentation the Schlichenmaiers conclusion is firstly tested, investigating the evolution of four different ARs. Hundreds of Intensitygram images from the Helioseismic and Magnetic Imager (HMI) images are used, obtained by the Solar Dynamics Observatory, in order to describe the area evolution of the above ARs and estimate the increase and decrease rates for umbra and penumbra areas, separately. A simple magnetohydrodynamic model is then tentatively used in a first approximation to explain the observed results.
Within the geometrical framework developed in arXiv:0705.2362, the problem of minimality for constrained calculus of variations is analysed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and are then reinterpreted in terms of Jacobi fields.
In this paper the first and second domain variation for functionals related to elliptic boundary and eigenvalue problems with Robin boundary conditions is computed. Minimality and maximality properties of the ball among nearly circular domains of given volume are derived. The discussion leads to the investigation of the eigenvalues of a Steklov eigenvalue problem. As a byproduct a general characterization of the optimal shapes is obtained.
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