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      36 views

      How many singular values are there for this function on Stiefel manifold

      Let $\mathcal{M}=\{X\in\mathbb{R}^{n\times r}: X^TX=I\}$ and define a function $f$ on $\mathcal{M}$: $$f(X)=\sum_{i=1}^n\left(\sum_{j=1}^rv_jX_{ij}^2\right)^2$$ where $(v_1,...,v_r)$ is some unit ...
      2
      votes
      0answers
      85 views

      Non-negative homogeneous polynomial on Stiefel manifold

      Let $\mathcal{M}=\{X\in\mathbb{R}^{n\times r}: X^TX=I\}$ where $r<n$ be the Stiefel manifold. Let $f: \mathcal{M}\to[0,1)$ be a homogeneous polynomial of the entries of $X=(X_{ij})_{1\leq i\leq n,...
      1
      vote
      1answer
      138 views

      A marginal space splitting $\{ \psi \}^{\perp}$

      Let $\psi \in L^2(\mathbb R^2,\mathbb C)$. Is there a continuous projection from $\{ \psi \}^{\perp}$ onto $$ \left\{ \varphi \in L^2(\mathbb R^2) \:\:\Big| \int \overline{\psi}(x,y) \varphi(x,y)\...
      3
      votes
      0answers
      53 views

      Pseudodifferential operator associated to a self-adjoint extension of a symmetric operator on an incomplete manifold

      Let $D$ be the Dirac operator acting on a spinor bundle $S$ over a complete Riemannian manifold $M$. Then $D$ is an essentially self-adjoint operator on $L^2(S)$. Suppose there is a compact subset $K\...
      3
      votes
      1answer
      133 views

      Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds

      On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
      0
      votes
      0answers
      66 views

      Compact embedding of the $\mathcal{C}^k$ norm on a compact Kahler manifold

      Given a smooth complex valued function $f$ on a Kahler manifold $X$, we can define its $\mathcal{C}^k$ norm to be $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, ...
      4
      votes
      1answer
      129 views

      Eigenvalues and Domain of the Laplace-Beltrami Operator

      Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...
      1
      vote
      0answers
      54 views

      Regularity for Laplacian operator on non-compact manifold

      Let $(M,g)$ be a complete non-compact Riemannian manifold . Thanks to @EveryLT, we know that the Poisson equation $$\Delta u=f,$$ is solvable for some $f\in L^2_k(M)$. Q Suppose that $(M,g)$ is ...
      4
      votes
      1answer
      144 views

      Orientability of moduli space and determinant bundle of ASD operator

      Setting In instanton gauge theory, given a $G$-principal bundle $P\to X^4$, the orientability of the moduli space of ASD connections $$\mathcal{M}_k = \{A \in L^{2}_{k}(X, \Lambda^1 \otimes\mathrm{...
      1
      vote
      0answers
      112 views

      Is this integral zero?

      I'd like to know if one integral expression I have can be shown to be zero for all possible cases. Let me introduce some notation. Consider $\mathfrak{g}=C^{\infty}(M)$ and the dual $\mathfrak{g}^*=\...
      2
      votes
      0answers
      105 views

      Euler-Lagrange equations on a differentiable manifold

      I am following the conventions of https://arxiv.org/abs/math-ph/9902027 Let $M$ be a differentiable manifold, $E \to M$ a vector bundle over $M$ with fibre $F$, $J^1(E)$ the rank-one jet bundle over $...
      1
      vote
      0answers
      44 views

      Derivation of the vortex filament equation from Euler equation

      How can the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, be derived from the Euler equation $$\partial_t \...
      3
      votes
      0answers
      43 views

      Controlling a Schwartz kernel near the diagonal

      Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...
      2
      votes
      0answers
      79 views

      Covariant derivative of the Monge-Ampere equation on Kähler manifolds

      I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
      8
      votes
      1answer
      258 views

      Laplacian spectrum asymptotics in neck stretching

      Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....

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