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Such objects solve an overdetermined elliptic free boundary problem in an exterior domain, with an additional boundary condition involving curvature and the Neumann trace. Motivated by a conjecture of Crowdy and Wegmann, we prove that every solution prescribed by a Jordan curve is circular for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$0 \\le \\textrm{We} \\le 3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mtext>We<\/mml:mtext>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , where 3 is the first bifurcation value. This range is sharp: the bifurcation branch at\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\textrm{We}=3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mtext>We<\/mml:mtext>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    contains non-circular solutions with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\textrm{We} &gt; 3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mtext>We<\/mml:mtext>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    arbitrarily close to 3. This elliptic problem describes critical points of the sum of the perimeter and the logarithmic potential energy of compact sets. We prove an area-constrained perimeter\u2013capacity inequality and show that, in the associated convexity-constrained variational problem, the unit disc is the unique minimiser up to translation if and only if\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$0 \\le \\textrm{We} \\le 3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mtext>We<\/mml:mtext>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>","DOI":"10.1007\/s00205-026-02262-9","type":"journal-article","created":{"date-parts":[[2026,10,22]],"date-time":"2026-10-22T08:11:40Z","timestamp":1792656700000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Global Rigidity of Two-Dimensional Bubbles"],"prefix":"10.1007","volume":"250","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3776-7697","authenticated-orcid":false,"given":"Lukas","family":"Niebel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,10,22]]},"reference":[{"key":"2262_CR1","unstructured":"Ahlfors, L.: Complex analysis\u2013an introduction to the theory of analytic functions of one complex variable. 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