Analysis And Spectral Theory Of Neck-Stretching Problems

Abstract We study the mapping properties of a large class of elliptic operators $$P_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> in gluing problems where two non-compact manifolds with asymptotically cylindrical geometry are glued along a neck of length 2 T . In the limit where $$T \rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , we reduce the question of constructing approximate solutions of $$P_T u = f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>T</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> to a finite-dimensional linear system, and provide a geometric interpretation of the obstructions to solving this system. Under some assumptions on the real roots of the model operator $$P_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> on the cylinder, we construct a Fredholm inverse for $$P_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> with good control on the growth of its norm. As applications of our method, we study the decay rate and density of the low eigenvalues of the Laplacian acting on differential forms, and give improved estimates for compact $$G_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -manifolds constructed by twisted connected sum. We relate our results to the swampland distance conjectures in physics.