Semi-Local Exotic Lagrangian Tori in Dimension Four

We study exotic Lagrangian tori in dimension four. In certain Stein domains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>p</mml:mi> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> </mml:math> (which naturally appear in almost toric fibrations) we find <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> families of monotone Lagrangian tori which are mutually distinct, up to symplectomorphisms. We prove that these remain distinct under embeddings of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>p</mml:mi> <mml:mi>q</mml:mi> </mml:mrow> </mml:msub> </mml:math> into geometrically bounded symplectic four-manifolds. We show that there are infinitely many different such embeddings when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is compact and (almost) toric and hence conclude that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> contains arbitrarily many Lagrangian tori which are distinct up to symplectomorphisms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> . In dimension four arbitrarily many different Lagrangian tori were previously known only in del Pezzo surfaces. Neither the embedded tori, nor the ambient space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> needs to be monotone for our methods to work.