Vertical curves and vertical fibers in the Heisenberg group

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the three-dimensional Heisenberg group. In this paper, we study vertical curves in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and fibers of maps <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H right-arrow double-struck upper R squared"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mo stretchy="false"> → </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {H}\to \mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from a metric perspective. We say that a set in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a vertical curve if it satisfies a cone condition with respect to a homogeneous cone with axis <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper Z mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:mi>Z</mml:mi> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle Z \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the center of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This is analogous to the cone condition used to define intrinsic Lipschitz graphs. In the first part of the paper, we prove that connected vertical curves are locally bi-Hölder equivalent to intervals. We also show that the class of vertical curves coincides with the class of intersections of intrinsic Lipschitz graphs satisfying a transversality condition. Unlike intrinsic Lipschitz graphs, the Hausdorff dimension of a vertical curve can vary; we construct vertical curves with Hausdorff dimension either strictly larger or strictly smaller than 2. Consequently, there are intersections of intrinsic Lipschitz graphs with Hausdorff dimension either strictly larger or strictly smaller than 2. In the second part of the paper, we consider smooth functions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi> β </mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from the unit ball <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {H}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that, in contrast to the situation in Euclidean space, there are maps such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"> <mml:semantics> <mml:mi> β </mml:mi> <mml:annotation encoding="application/x-tex">\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is arbitrarily close to the projection <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {H}</mml:annotation>