Minimal surfaces with symmetries
IF 1.5
1区 数学
Q1 MATHEMATICS
Franc Forstnerič
{"title":"Minimal surfaces with symmetries","authors":"Franc Forstnerič","doi":"10.1112/plms.12590","DOIUrl":null,"url":null,"abstract":"Let <mjx-container aria-label=\"upper G\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/7c714f1a-0309-4565-b8ca-97fcb85334f7/plms12590-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\">G</mi>$G$</annotation></semantics></math></mjx-assistive-mml></mjx-container> be a finite group acting on a connected open Riemann surface <mjx-container aria-label=\"upper X\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/6e2516dd-8e0d-45a2-933f-22437e1a1173/plms12590-math-0002.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\">X</mi>$X$</annotation></semantics></math></mjx-assistive-mml></mjx-container> by holomorphic automorphisms and acting on a Euclidean space <mjx-container aria-label=\"double struck upper R Superscript n\" ctxtmenu_counter=\"2\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-msup data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"numbersetletter\" data-semantic-speech=\"double struck upper R Superscript n\" data-semantic-type=\"superscript\"><mjx-mi data-semantic-font=\"double-struck\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/1f687343-abd4-4291-8ba7-5b1478b9c67a/plms12590-math-0003.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><msup data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"numbersetletter\" data-semantic-speech=\"double struck upper R Superscript n\" data-semantic-type=\"superscript\"><mi data-semantic-=\"\" data-semantic-font=\"double-struck\" data-semantic-parent=\"2\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"identifier\" mathvariant=\"double-struck\">R</mi><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></msup>$\\mathbb {R}^n$</annotation></semantics></math></mjx-assistive-mml></mjx-container> <mjx-container aria-label=\"left parenthesis n greater than or slanted equals 3 right parenthesis\" ctxtmenu_counter=\"3\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"5\" data-semantic-content=\"0,4\" data-semantic- data-semantic-role=\"leftright\" data-semantic-speech=\"left parenthesis n greater than or slanted equals 3 right parenthesis\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"1,3\" data-semantic-content=\"2\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"inequality\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,⩾\" data-semantic-parent=\"5\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn></mjx-mrow><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/1f06f45e-4243-46c3-8b7c-89a14ac340e4/plms12590-math-0004.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"5\" data-semantic-content=\"0,4\" data-semantic-role=\"leftright\" data-semantic-speech=\"left parenthesis n greater than or slanted equals 3 right parenthesis\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semantic-type=\"fence\" stretchy=\"false\">(</mo><mrow data-semantic-=\"\" data-semantic-children=\"1,3\" data-semantic-content=\"2\" data-semantic-parent=\"6\" data-semantic-role=\"inequality\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"5\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,⩾\" data-semantic-parent=\"5\" data-semantic-role=\"inequality\" data-semantic-type=\"relation\">⩾</mo><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"5\" data-semantic-role=\"integer\" data-semantic-type=\"number\">3</mn></mrow><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"close\" data-semantic-type=\"fence\" stretchy=\"false\">)</mo></mrow>$(n\\geqslant 3)$</annotation></semantics></math></mjx-assistive-mml></mjx-container> by orthogonal transformations. We identify a necessary and sufficient condition for the existence of a <mjx-container aria-label=\"upper G\" ctxtmenu_counter=\"4\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/4db0de79-1d5f-4c89-96ed-9612c03fe5f3/plms12590-math-0005.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\">G</mi>$G$</annotation></semantics></math></mjx-assistive-mml></mjx-container>-equivariant conformal minimal immersion <mjx-container aria-label=\"upper F colon upper X right arrow double struck upper R Superscript n\" ctxtmenu_counter=\"5\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,1,7\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"sequence\" data-semantic-speech=\"upper F colon upper X right arrow double struck upper R Superscript n\" data-semantic-type=\"punctuated\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"punctuated\" data-semantic-parent=\"8\" data-semantic-role=\"colon\" data-semantic-type=\"punctuation\" rspace=\"2\" space=\"1\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"2,6\" data-semantic-content=\"3\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"arrow\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,→\" data-semantic-parent=\"7\" data-semantic-role=\"arrow\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-msup data-semantic-children=\"4,5\" data-semantic- data-semantic-parent=\"7\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"superscript\"><mjx-mi data-semantic-font=\"double-struck\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/04ecbfec-7513-43e3-9fea-e6b0eea50a7f/plms12590-math-0006.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,1,7\" data-semantic-content=\"1\" data-semantic-role=\"sequence\" data-semantic-speech=\"upper F colon upper X right arrow double struck upper R Superscript n\" data-semantic-type=\"punctuated\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"8\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">F</mi><mo data-semantic-=\"\" data-semantic-operator=\"punctuated\" data-semantic-parent=\"8\" data-semantic-role=\"colon\" data-semantic-type=\"punctuation\">:</mo><mrow data-semantic-=\"\" data-semantic-children=\"2,6\" data-semantic-content=\"3\" data-semantic-parent=\"8\" data-semantic-role=\"arrow\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"7\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">X</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,→\" data-semantic-parent=\"7\" data-semantic-role=\"arrow\" data-semantic-type=\"relation\">→</mo><msup data-semantic-=\"\" data-semantic-children=\"4,5\" data-semantic-parent=\"7\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"superscript\"><mi data-semantic-=\"\" data-semantic-font=\"double-struck\" data-semantic-parent=\"6\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"identifier\" mathvariant=\"double-struck\">R</mi><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></msup></mrow></mrow>$F:X\\rightarrow \\mathbb {R}^n$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. We show in particular that such a map <mjx-container aria-label=\"upper F\" ctxtmenu_counter=\"6\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper F\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/c1e30311-1482-4eb5-aa82-fe2e51dad5d7/plms12590-math-0007.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper F\" data-semantic-type=\"identifier\">F</mi>$F$</annotation></semantics></math></mjx-assistive-mml></mjx-container> always exists if <mjx-container aria-label=\"upper G\" ctxtmenu_counter=\"7\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/88e06008-f094-4357-b71b-d4a65a7b83cb/plms12590-math-0008.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\">G</mi>$G$</annotation></semantics></math></mjx-assistive-mml></mjx-container> acts without fixed points on <mjx-container aria-label=\"upper X\" ctxtmenu_counter=\"8\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/5dd62454-0521-4a28-aae3-c7446cbec6d8/plms12590-math-0009.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\">X</mi>$X$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. Furthermore, every finite group <mjx-container aria-label=\"upper G\" ctxtmenu_counter=\"9\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/50c32250-ef41-4eb3-95b5-1d3292e2e79f/plms12590-math-0010.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper G\" data-semantic-type=\"identifier\">G</mi>$G$</annotation></semantics></math></mjx-assistive-mml></mjx-container> arises in this way for some open Riemann surface and <mjx-container aria-label=\"n equals 2 StartAbsoluteValue upper G EndAbsoluteValue\" ctxtmenu_counter=\"10\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-children=\"0,8\" data-semantic-content=\"1\" data-semantic- data-semantic-role=\"equality\" data-semantic-speech=\"n equals 2 StartAbsoluteValue upper G EndAbsoluteValue\" data-semantic-type=\"relseq\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"relseq,=\" data-semantic-parent=\"9\" data-semantic-role=\"equality\" data-semantic-type=\"relation\" rspace=\"5\" space=\"5\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"2,6\" data-semantic-content=\"7\" data-semantic- data-semantic-parent=\"9\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"integer\" data-semantic-type=\"number\"><mjx-c></mjx-c></mjx-mn><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"infixop,\" data-semantic-parent=\"8\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"neutral\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"neutral\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"neutral\" data-semantic-type=\"fence\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-mrow></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/4dbcae11-f627-4a38-bad2-ab571411660b/plms12590-math-0011.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow data-semantic-=\"\" data-semantic-children=\"0,8\" data-semantic-content=\"1\" data-semantic-role=\"equality\" data-semantic-speech=\"n equals 2 StartAbsoluteValue upper G EndAbsoluteValue\" data-semantic-type=\"relseq\"><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"9\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi><mo data-semantic-=\"\" data-semantic-operator=\"relseq,=\" data-semantic-parent=\"9\" data-semantic-role=\"equality\" data-semantic-type=\"relation\">=</mo><mrow data-semantic-=\"\" data-semantic-annotation=\"clearspeak:unit\" data-semantic-children=\"2,6\" data-semantic-content=\"7\" data-semantic-parent=\"9\" data-semantic-role=\"implicit\" data-semantic-type=\"infixop\"><mn data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic-parent=\"8\" data-semantic-role=\"integer\" data-semantic-type=\"number\">2</mn><mo data-semantic-=\"\" data-semantic-added=\"true\" data-semantic-operator=\"infixop,\" data-semantic-parent=\"8\" data-semantic-role=\"multiplication\" data-semantic-type=\"operator\"></mo><mrow data-semantic-=\"\" data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic-parent=\"8\" data-semantic-role=\"neutral\" data-semantic-type=\"fenced\"><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"neutral\" data-semantic-type=\"fence\" stretchy=\"false\">|</mo><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"6\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">G</mi><mo data-semantic-=\"\" data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"neutral\" data-semantic-type=\"fence\" stretchy=\"false\">|</mo></mrow></mrow></mrow>$n=2|G|$</annotation></semantics></math></mjx-assistive-mml></mjx-container>. We obtain an analogous result for minimal surfaces having complete ends with finite total Gaussian curvature, and for discrete groups acting on <mjx-container aria-label=\"upper X\" ctxtmenu_counter=\"11\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/3b365afb-f31b-4887-9434-dd38fd67c43b/plms12590-math-0012.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\">X</mi>$X$</annotation></semantics></math></mjx-assistive-mml></mjx-container> properly discontinuously and acting on <mjx-container aria-label=\"double struck upper R Superscript n\" ctxtmenu_counter=\"12\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-msup data-semantic-children=\"0,1\" data-semantic- data-semantic-role=\"numbersetletter\" data-semantic-speech=\"double struck upper R Superscript n\" data-semantic-type=\"superscript\"><mjx-mi data-semantic-font=\"double-struck\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: 0.363em;\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\" size=\"s\"><mjx-c></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/b80df117-6698-4b7d-9483-423a6011ffd0/plms12590-math-0013.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><msup data-semantic-=\"\" data-semantic-children=\"0,1\" data-semantic-role=\"numbersetletter\" data-semantic-speech=\"double struck upper R Superscript n\" data-semantic-type=\"superscript\"><mi data-semantic-=\"\" data-semantic-font=\"double-struck\" data-semantic-parent=\"2\" data-semantic-role=\"numbersetletter\" data-semantic-type=\"identifier\" mathvariant=\"double-struck\">R</mi><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-parent=\"2\" data-semantic-role=\"latinletter\" data-semantic-type=\"identifier\">n</mi></msup>$\\mathbb {R}^n$</annotation></semantics></math></mjx-assistive-mml></mjx-container> by rigid transformations.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"40 1","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/plms.12590","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let具有对称性的最小曲面
让 G$G$ 是一个有限群，通过全形自变量作用于一个连通的开黎曼曲面 X$X$，并通过正交变换作用于欧几里得空间 Rn$\mathbb {R}^n$ (n⩾3)$(n\geqslant 3)$。我们确定了 G$G$ 传递共形最小浸入 F:X→Rn$F:X\rightarrow \mathbb {R}^n$ 存在的必要条件和充分条件。我们特别证明，如果 G$G$ 在 X$X$ 上无定点作用，那么这样的映射 F$F$ 总是存在的。此外，对于某些开放黎曼曲面和 n=2|G|$n=2|G|$，每个有限群 G$G$ 都是这样产生的。对于具有有限总高斯曲率的完整端点的极小曲面，以及通过刚性变换作用于 X$X$ 的离散群，以及作用于 Rn$\mathbb {R}^n$ 的离散群，我们得到了类似的结果。
本文章由计算机程序翻译，如有差异，请以英文原文为准。
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来源期刊
CiteScore
2.90
自引率
0.00%
发文量
82
审稿时长
6-12 weeks
期刊介绍：
The Proceedings of the London Mathematical Society is the flagship journal of the LMS. It publishes articles of the highest quality and significance across a broad range of mathematics. There are no page length restrictions for submitted papers.
The Proceedings has its own Editorial Board separate from that of the Journal, Bulletin and Transactions of the LMS.
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